Explanations

Where does the phrase "flipping someone the bird" come from?

I couldn't find anything on the use of "bird" for finger in this phrase.

The following, from Eric Partridge's A Dictionary of Slang and Unconventional English may be relevant although it makes no mention of the hand gesture. To give someone the bird is "to dismiss [him], send him about his business . . . late C. 19-20. [From] the theatre . . . In Australia, 'give the bird' is to treat with derision: from before 1916."
In obsolete theatrical usage (Partridge gives a date of 1883), "the bird" is defined as "a hissing of an actor," from the sound made by geese.

I'm wondering whether we can point the finger at Cockney rhyming slang again here... always a useful last resort. Could "bird" be rhyming slang for "third", as in third finger? Or is this too far-fetched?

Being far-fetched or not isn't an important criterion. Sometimes the correct explanation is less intuitively plausible than the incorrect ones. Maybe "bird" originated as rhyming slang, maybe not--what we need is some historical evidence.