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I get the feeling this trope should be in the wiki, but if there's a defined name for it then I don't know what it is.
Alice and Bob are separating for the last time. The cause can vary from a Heroic Sacrifice to simply a break up, it doesn't really matter - the point is that they're leaving each other and they probably won't see one another again (or at least, that's how it's played out for the audience). They both probably have tons of things to say, but instead of giving a Final Speech, one (or both) of them just say, "Goodbye." In other words, it's a final goodbye.
A possible subversion I've also seen is that the characters replace "goodbye" with "thank you," because saying "goodbye" gives the scene a feeling of finality that neither wants to acknowledge.
It isn't quite the same as Famous Last Words, since that usually involves at least one sentence and is spoken before someone dies. Death isn't a requirement in this case.
Edited by UnluckyNormalcyCLO