The first place where the amenability problem for Thompson's group $F$ appears in the literature is, I believe, 1980 in a problems article by Ross Geoghegan. I have heard, however, vague comments to the effect that the problem was considered by other people before this. Does anyone have any knowledge about the existence of this problem prior to 1980?
Edit: Following Mark's advice offline, I wrote Richard Thompson to verify the details of Mark's answer. He did confirm that he considered the problem. He first observed that his group $F$ did not contain $\mathbb{F}_2$. He then discovered the material on amenability in Hewitt and Ross's text on abstract harmonic analysis. He then observed that $F$ was not elementarily amenable. This occurred sometime prior to his 1973 visit to University of Illinois at Urbana-Champaign to visit Day.
He did not, however, attend Greenleaf's series of lectures in 1967. Instead he read Greenleaf's 1969 text "Invariant Means on Topological Groups and Their Applications." It seems that the best date to attach to Thompson's consideration of the question is 1973. Of course Thompson never published his observations and they were not widely circulated. His observations mentioned above were rediscovered by others in the 1980s (the question itself by Geoghegan).
I have invited Richard to post an answer, in which case I will delete this edit.
Best Answer
Richard Thompson visited me at Princeton several times I believe in the mid '70's, gave me copies of some handwritten notes about his groups (which I shared with a few people), and raised the amenability question, which we discussed a bit. I don't think Thompson's groups were very widely known at the time.