I would use IsolatedStorageSettings to store the list and compare the dynamic text to the list in the isolatedstoragesettings upon button click. Then on FavouritesList page, set itemsource of the listbox to the list in IsolatedStorageSettings.So here are the steps to be followed:
1. Create a model/class to set the dynamic text being shown on the text block   
public class favourites
{
    public string myText { get; set; }
}
2. In the button click event on MainPage.xaml.cs, first set the dynamic text (where ever you are getting it from) to the text block if you need to and then create the list and/or compare
 private void AddToFavoritesButton_Click(object sender, RoutedEventArgs e)
    {
        //your dynamic text set to textblock
        StringTextBlock.Text = myDynamicText;  
        //Set value of your text to member variable of the model/class
        favourites f = new favourites();
        f.myText = myDynamicText;
        IsolatedStorageSettings settings = IsolatedStorageSettings.ApplicationSettings;
        /*Check if "FavouritesList" key is present in IsolatedStorageSettings
          which means already a list had been added. If yes, retrieve the
          list, compare each item with your dynamic text, add or remove
          accordingly and replace the new list in IsolatedStorageSettings
          with same key. */
        if (settings.Contains("FavouritesList"))
        {
            List<favourites> l = (List<favourites>)settings["FavouritesList"];
            for(int i = 0; i <= l.Count()-1; i++)
            {
                if (l[i].Equals(myDynamicText))
                {
                    l.RemoveAt(i);
                    settings["FavouritesList"] = l;
                }
                else
                {
                    l.Add(f);
                    settings["FavouritesList"] = l;
                }
            }           
        }
        //If no key in IsolatedStorageSettings means no data has been added
        //in list and IsolatedStorageSettings. So add new data
        else
        {
            List<favourites> l = new List<favourites>();
            l.Add(f);
            settings["FavouritesList"] = l;
        }
        settings.Save();
    }       
Now all that is left is show the always updated list in the FavouritesList Page. I added a 'NoData' textblock that should be visible when there is nothing in the list. Else the list will be displayed.
In FavouritesList.xaml
 <ListBox x:Name="FavoriteListBox" Visibility="Collapsed">
            <ListBox.ItemTemplate>
                <DataTemplate>
                    <TextBlock Text="{Binding myText}"/>
                </DataTemplate>
            </ListBox.ItemTemplate>
        </ListBox>
        <TextBlock Name="NoData" 
                   Text="No Data" 
                   Visibility="Collapsed" 
                   Width="50" 
                   Height="50"/>
In FavouritesList.xaml.cs
 IsolatedStorageSettings settings = IsolatedStorageSettings.ApplicationSettings;
        if (settings.Contains("FavouritesList"))
        {
           List<favourites> l = (List<favourites>)settings["FavouritesList"];
            if(l.Count!= 0)
            {
                NoData.Visibility = System.Windows.Visibility.Collapsed;
                FavoriteListBox.Visibility = System.Windows.Visibility.Visible;
                FavoriteListBox.ItemsSource = l;
            }                  
        }
        else
        {
            FavoriteListBox.Visibility = System.Windows.Visibility.Collapsed;                   
            NoData.Visibility = System.Windows.Visibility.Visible;
        }
I have not tested this but should definitely work. Hope it helps!