In my ACKS game the rule is that you roll for hit points on first level. No kickers, no “max hit die on first level”, no rerolls.
I don’t think playing a 1hp character on first level is that big a deal — quite the opposite, it’s awesome when you survive 1st level. However, there’s a side-effect I’ve experienced as a player that I don’t like at all.
I once played a magic-user who rolled 1 on the 1d4 for first level. That’s 1:4 chance. Then I rolled 1 for second level, for a cumulative chance of 1:16. When third level came up I rolled 1 again, which made for a 1:64 cumulative chance — so even though it sucked, it wasn’t even that unlikely. What really irked me about it was how those rolls carried irrevocably in to the future of the character. “Roll badly, and you’ll suffer for it forever.”
I still like rolling hit points, though.
So, a house rule: when leveling up you can either (1) add 1 HD + CON modifier hit points as usual, or you can (2) reroll all your hit dice, adding your CON modifier for each — and using the new score only if it’s better than the old one.
Using this rule for my magic user I would have rolled 2d4 for second level, which would have made having 2hp on level two a 1:64 chance, and 3d4 for level three would have made ending up with 3hp a cumulative 1:4096 chance. It could still have happened, but with odds like that at least I could have righteously felt snubbed by the gods.
So, there are no guarantees, but if you roll badly you get to reroll that hit die in the future — which biases the results against lowball results.
A big softie I am. Coddle players I do.