Would it be OK to have a single instance of SQLiteOpenHelper as a member of a subclassed Application, and have all Activities that need an instance of SQLiteDatabase get it from the one helper?
Having a single SQLiteOpenHelper instance can help in threading cases. Since all threads would share the common SQLiteDatabase, synchronization of operations is provided.
However, I wouldn't make a subclass of Application. Just have a static data member that is your SQLiteOpenHelper. Both approaches give you something accessible from anywhere. However, there is only one subclass of Application, making it more difficult for you to use other subclasses of Application (e.g., GreenDroid requires one IIRC). Using a static data member avoids that. However, do use the Application Context when instantiating this static SQLiteOpenHelper (constructor parameter), so you do not leak some other Context.
And, in cases where you aren't dealing with multiple threads, you can avoid any possible memory leak issues by just using one SQLiteOpenHelper instance per component. However, in practice, you should be dealing with multiple threads (e.g., a Loader), so this recommendation is only relevant for trivial applications, such as those found in some books... :-)