Some discussions of the movie reference a peaceful, almost zen acceptance, as well as the idea that
The Dude will always be around.
A discussion on Reddit of what "the Dude abides" means has some consensus that it's an
Intentionally vague phrase hinting at the fact that The Dude Lives, in his unperturbable state of dudeness, somewhere.
and that the definitions "accept" and "continue" make sense in this context.
As @wfaulk points out, today we usually use abide transitively to mean things like trusting in, accepting or obeying; so it doesn't mean The Dude accepts or endures a particular thing, but I agree that the phrase still can imply a sense of patience or toleration, even if it's just the way someone waits or continues.
If you look at the etymology of abide, you can see how some of these meanings emerge:
O.E. abidan, gebidan "remain, wait, delay, remain behind," from ge- completive prefix (denoting onward motion; see a- (1)) + bidan "bide, remain, wait, dwell" (see bide). ... Meaning "to put up with" (now usually negative) first recorded 1520s.
and going back to bide:
O.E. bidan "to stay, continue, live, remain," also "to trust, rely" (cognate of O.N. biða, O.Fris. bidia, Goth. beidan "to wait"), apparently from PIE *bheidh-, an extended stem of one root of O.E. biddan (see bid), the original sense of which was "to command," and "to trust" (cf. Gk. peithein "to persuade," pistis "faith;" L. fidere "to trust," foedus "compact, treaty," O.C.S. beda "need"). Perhaps the sense evolved in prehistoric times through "endure," and "endure a wait," to "to wait."
I think you're right: The Dude endures; The Dude lives on.