Those how have an absolute pitch can name the note when they hear it. For those who don't have this ability is it possible to remember a pitch of at least one note?
Yes, it seems like this ability can be acquired with practice.
In Identifying Notes without Perfect Pitch, the author picked twelve familiar songs, each of which started on one of the twelve pitches of the chromatic scale. Once he memorized the song-pitch association, he was able to recognize 10 out of of 12 pitches.
Some users of Music Stack Exchange mention similar results. For example, one person says they can remember the pitch of their tuning fork, and another suggests the song approach: "If hearing a note instantly reminds you of a song, then I think you are identifying that note, just by the song name, not the note name (that you may or may not happen to know)."