Fixed amount. For three reasons:
a) Damage is random already - but averages out, hit points don't, rolling low hit points is equivalent to giving all enemies loaded damage dice against you.
b) Hit point rolling is always done with a single dice, meaning the probability is flat... if it was a bell-curve like probability I'd have less problems with it.
c) It's a lot easier to balance a game around a less wildly varying hit point value (where a fighter 3 in one game has 12 hit points in one game and a fighter 2 has 20 hit points in another).
So vast preference for fixed as default option (that players always should have). I don't mind giving people the option to roll though, if they really want to - especially if it's then a "hybrid roll" like 1d4+2 etc.
EDIT: I totally see that as prime material for a module, though. Because if you're going to a long campaign, possibly story-driven, having the potential to cripple a character without making a single bad choice is annoying. If it's a campaign where I'll burn through several characters in a shorter period (e.g. a beer & pretzels dungeon crawl campaign), then the tactical challenge can be interesting - but then mark it as a "hardcore crawl module".