It turns out this is possible without changing the source of osmdroid, due to the public MapView(Context context, int tileSizePixels, ResourceProxy resourceProxy, MapTileProviderBase aTileProvider) constrtuctor.
Assuming you already have a custom class like MySSLSocketFactory (which extends javax.net.ssl.SSLSocketFactory), the basic process looks like this:
- Create a drop-in replacement class for - MapTileDownloaderto perform the download in a way that makes use of- MySSLSocketFactory. Let's call this- MyTileDownloader.
 
- Create a drop-in replacement class for - MapTileProviderBasicthat instantiates your custom- MyTileDownloader. Let's call this- MyTileProvider.
 
- Instantiate your tile source as a - new XYTileSource(no need to write a custom class).
 
- Instantiate - MyTileProviderwith your tile source instance.
 
- Instantiate - MapVewwith your tile provider instance.
 
MySSLSocketFactory is left as an exercise for the reader. See this post.
MyTileDownloader looks something like this:
import java.io.BufferedInputStream;
import java.io.BufferedOutputStream;
import java.io.ByteArrayInputStream;
import java.io.ByteArrayOutputStream;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.net.HttpURLConnection;
import java.net.URL;
import java.net.UnknownHostException;
import javax.net.ssl.HttpsURLConnection;
import javax.net.ssl.SSLSocketFactory;
import org.osmdroid.tileprovider.MapTile;
import org.osmdroid.tileprovider.MapTileRequestState;
import org.osmdroid.tileprovider.modules.IFilesystemCache;
import org.osmdroid.tileprovider.modules.INetworkAvailablityCheck;
import org.osmdroid.tileprovider.modules.MapTileDownloader;
import org.osmdroid.tileprovider.modules.MapTileModuleProviderBase;
import org.osmdroid.tileprovider.tilesource.BitmapTileSourceBase.LowMemoryException;
import org.osmdroid.tileprovider.tilesource.ITileSource;
import org.osmdroid.tileprovider.tilesource.OnlineTileSourceBase;
import org.osmdroid.tileprovider.util.StreamUtils;
import android.graphics.drawable.Drawable;
import android.text.TextUtils;
import android.util.Log;
/**
 * A drop-in replacement for {@link MapTileDownloader}. This loads tiles from an
 * HTTP or HTTPS server, making use of a custom {@link SSLSocketFactory} for SSL
 * peer verification.
 */
public class MyTileDownloader extends MapTileModuleProviderBase {
    private static final String TAG = "MyMapTileDownloader";
    protected OnlineTileSourceBase mTileSource;
    protected final IFilesystemCache mFilesystemCache;
    protected final INetworkAvailablityCheck mNetworkAvailablityCheck;
    protected final SSLSocketFactory mSSLSocketFactory;
    public MyTileDownloader(ITileSource pTileSource,
            IFilesystemCache pFilesystemCache,
            INetworkAvailablityCheck pNetworkAvailablityCheck,
            SSLSocketFactory pSSLSocketFactory) {
        super(4, TILE_DOWNLOAD_MAXIMUM_QUEUE_SIZE);
        setTileSource(pTileSource);
        mFilesystemCache = pFilesystemCache;
        mNetworkAvailablityCheck = pNetworkAvailablityCheck;
        mSSLSocketFactory = pSSLSocketFactory;
    }
    public ITileSource getTileSource() {
        return mTileSource;
    }
    @Override
    public void setTileSource(final ITileSource tileSource) {
        // We are only interested in OnlineTileSourceBase tile sources
        if (tileSource instanceof OnlineTileSourceBase)
            mTileSource = (OnlineTileSourceBase) tileSource;
        else
            mTileSource = null;
    }
    @Override
    public boolean getUsesDataConnection() {
        return true;
    }
    @Override
    protected String getName() {
        return "Online Tile Download Provider";
    }
    @Override
    protected String getThreadGroupName() {
        return "downloader";
    }
    @Override
    public int getMinimumZoomLevel() {
        return (mTileSource != null ? mTileSource.getMinimumZoomLevel()
                : MINIMUM_ZOOMLEVEL);
    }
    @Override
    public int getMaximumZoomLevel() {
        return (mTileSource != null ? mTileSource.getMaximumZoomLevel()
                : MAXIMUM_ZOOMLEVEL);
    }
    @Override
    protected Runnable getTileLoader() {
        return new TileLoader();
    };
    private class TileLoader extends MapTileModuleProviderBase.TileLoader {
        @Override
        public Drawable loadTile(final MapTileRequestState aState)
                throws CantContinueException {
            if (mTileSource == null)
                return null;
            InputStream in = null;
            OutputStream out = null;
            final MapTile tile = aState.getMapTile();
            try {
                if (mNetworkAvailablityCheck != null
                        && !mNetworkAvailablityCheck.getNetworkAvailable()) {
                    if (DEBUGMODE)
                        Log.d(TAG, "Skipping " + getName()
                                + " due to NetworkAvailabliltyCheck.");
                    return null;
                }
                final String tileURLString = mTileSource.getTileURLString(tile);
                if (DEBUGMODE)
                    Log.d(TAG, "Downloading Maptile from url: " + tileURLString);
                if (TextUtils.isEmpty(tileURLString))
                    return null;
                // Create an HttpURLConnection to download the tile
                URL url = new URL(tileURLString);
                HttpURLConnection connection = (HttpURLConnection) url
                        .openConnection();
                connection.setConnectTimeout(30000);
                connection.setReadTimeout(30000);
                // Use our custom SSLSocketFactory for secure connections
                if ("https".equalsIgnoreCase(url.getProtocol()))
                    ((HttpsURLConnection) connection)
                            .setSSLSocketFactory(mSSLSocketFactory);
                // Open the input stream
                in = new BufferedInputStream(connection.getInputStream(),
                        StreamUtils.IO_BUFFER_SIZE);
                // Check to see if we got success
                if (connection.getResponseCode() != 200) {
                    Log.w(TAG, "Problem downloading MapTile: " + tile
                            + " HTTP response: " + connection.getHeaderField(0));
                    return null;
                }
                // Read the tile into an in-memory byte array
                final ByteArrayOutputStream dataStream = new ByteArrayOutputStream();
                out = new BufferedOutputStream(dataStream,
                        StreamUtils.IO_BUFFER_SIZE);
                StreamUtils.copy(in, out);
                out.flush();
                final byte[] data = dataStream.toByteArray();
                final ByteArrayInputStream byteStream = new ByteArrayInputStream(
                        data);
                // Save the data to the filesystem cache
                if (mFilesystemCache != null) {
                    mFilesystemCache.saveFile(mTileSource, tile, byteStream);
                    byteStream.reset();
                }
                final Drawable result = mTileSource.getDrawable(byteStream);
                return result;
            } catch (final UnknownHostException e) {
                Log.w(TAG, "UnknownHostException downloading MapTile: " + tile
                        + " : " + e);
                throw new CantContinueException(e);
            } catch (final LowMemoryException e) {
                Log.w(TAG, "LowMemoryException downloading MapTile: " + tile
                        + " : " + e);
                throw new CantContinueException(e);
            } catch (final FileNotFoundException e) {
                Log.w(TAG, "Tile not found: " + tile + " : " + e);
            } catch (final IOException e) {
                Log.w(TAG, "IOException downloading MapTile: " + tile + " : "
                        + e);
            } catch (final Throwable e) {
                Log.e(TAG, "Error downloading MapTile: " + tile, e);
            } finally {
                StreamUtils.closeStream(in);
                StreamUtils.closeStream(out);
            }
            return null;
        }
        @Override
        protected void tileLoaded(final MapTileRequestState pState,
                final Drawable pDrawable) {
            // Don't return the tile Drawable because we'll wait for the fs
            // provider to ask for it. This prevent flickering when a load
            // of delayed downloads complete for tiles that we might not
            // even be interested in any more.
            super.tileLoaded(pState, null);
        }
    }
}
MyTileProvider looks something like this.
Note that you'll need a way to get access to your instance of MySSLSocketFactory inside this class. This is left as an exercise for the reader. I did this using app.getSSLSocketFactory(), where app is an instance of a custom class that extends Application, but your mileage may vary.
import javax.net.ssl.SSLSocketFactory;
import org.osmdroid.tileprovider.IMapTileProviderCallback;
import org.osmdroid.tileprovider.IRegisterReceiver;
import org.osmdroid.tileprovider.MapTileProviderArray;
import org.osmdroid.tileprovider.MapTileProviderBasic;
import org.osmdroid.tileprovider.modules.INetworkAvailablityCheck;
import org.osmdroid.tileprovider.modules.MapTileFileArchiveProvider;
import org.osmdroid.tileprovider.modules.MapTileFilesystemProvider;
import org.osmdroid.tileprovider.modules.NetworkAvailabliltyCheck;
import org.osmdroid.tileprovider.modules.TileWriter;
import org.osmdroid.tileprovider.tilesource.ITileSource;
import org.osmdroid.tileprovider.util.SimpleRegisterReceiver;
import android.content.Context;
/**
 * A drop-in replacement for {@link MapTileProviderBasic}. This top-level tile
 * provider implements a basic tile request chain which includes a
 * {@link MapTileFilesystemProvider} (a file-system cache), a
 * {@link MapTileFileArchiveProvider} (archive provider), and a
 * {@link MyTileDownloader} (downloads map tiles via tile source).
 */
public class MyTileProvider extends MapTileProviderArray implements
        IMapTileProviderCallback {
    public MyTileProvider(final Context pContext, final ITileSource pTileSource) {
        this(new SimpleRegisterReceiver(pContext),
                new NetworkAvailabliltyCheck(pContext), pTileSource, app
                        .getSSLSocketFactory());
    }
    protected MyTileProvider(final IRegisterReceiver pRegisterReceiver,
            final INetworkAvailablityCheck aNetworkAvailablityCheck,
            final ITileSource pTileSource,
            final SSLSocketFactory pSSLSocketFactory) {
        super(pTileSource, pRegisterReceiver);
        // Look for raw tiles on the file system
        final MapTileFilesystemProvider fileSystemProvider = new MapTileFilesystemProvider(
                pRegisterReceiver, pTileSource);
        mTileProviderList.add(fileSystemProvider);
        // Look for tile archives on the file system
        final MapTileFileArchiveProvider archiveProvider = new MapTileFileArchiveProvider(
                pRegisterReceiver, pTileSource);
        mTileProviderList.add(archiveProvider);
        // Look for raw tiles on the Internet
        final TileWriter tileWriter = new TileWriter();
        final MyTileDownloader downloaderProvider = new MyTileDownloader(
                pTileSource, tileWriter, aNetworkAvailablityCheck,
                pSSLSocketFactory);
        mTileProviderList.add(downloaderProvider);
    }
}
Finally, the instantiation looks something like this:
XYTileSource tileSource = new XYTileSource("MapQuest", null, 3, 8, 256, ".jpg",
    "https://10.0.0.1/path/to/your/map/tiles/");
MapTileProviderBase tileProvider = new MyTileProvider(context, tileSource);
ResourceProxy resourceProxy = new DefaultResourceProxyImpl(context);
MapView mapView = new MapView(context, 256, resourceProxy, tileProvider);