I am also bothered by this phrase, even though I have been known to use it. The problem for writers is that there are not many alternatives. You can ignore the connection between the place and the name on the assumption that it is obvious. Phrases like "explored the lake that is now named after him" are more awkward. "Eponymous river" sounds phony. None of the choices sound very good to me, although I agree that it time to give name bearing a rest.