Here's my two favorite ways...  
// Executing an anonymous script
function exec(fn) {
   var script = document.createElement('script');
   script.setAttribute("type", "application/javascript");
   script.textContent = '(' + fn + ')();';
   document.documentElement.appendChild(script); // run the script
   document.documentElement.removeChild(script); // clean up
}
script = function() {
//sayHello();
alert('hello');
}
exec(script);
// Append a script from a file in your extension
function appendScript(scriptFile) {
   var script = document.createElement('script');
   script.setAttribute("type", "application/javascript");
   script.setAttribute("src", chrome.extension.getURL(scriptFile));
   document.documentElement.appendChild(script); // run the script
}
appendScript('someFile.js');
Also chrome.tabs.executeScript() can be used from a browser/page action popup and the above code works in a content script aswell.
EDIT
Thanks to comments by @renocor, here's a variation of the first method that allows you to send arguments to the injected function....
function exec(fn) {
    var args = '';
    if (arguments.length > 1) {
        for (var i = 1, end = arguments.length - 2; i <= end; i++) {
            args += typeof arguments[i]=='function' ? arguments[i] : JSON.stringify(arguments[i]) + ', ';
        }
        args += typeof arguments[i]=='function' ? arguments[arguments.length - 1] : JSON.stringify(arguments[arguments.length - 1]);
    }
    var script = document.createElement('script');
    script.setAttribute("type", "application/javascript");
    script.textContent = '(' + fn + ')(' + args + ');';
    document.documentElement.appendChild(script); // run the script
    document.documentElement.removeChild(script); // clean up
}
script = function(what, huh, nah, yeah) {
    console.debug(arguments);
    console.debug('what=', what);
    console.debug('huh=', huh);
    console.debug('nah=', nah);
    console.debug('yeah=', yeah);
    if (typeof yeah=='function') yeah();
}
exec(script, 'meh', ['bleh'], {
    a: {
        b: 0
    }
}, function(){
    alert('hi');
});
console.debug('No arguments');
exec(script);