I tend to take average as a player, and give the choice to my players when I DM. In one game, we allowed a minimum on the Hit Die roll equal to the character's CON modifier. Until we get a 20 Con wizard that comletely breaks the average, it has worked ok...
TBH, I long for a way to add a bit of variance without the full range of the HD. Perhaps one dice category lower + 2? (e.g. 1d8 + Con becomes 1d6+2 + Con)