HP Rolling Alternatives

Not realizing this seems to be common, so I'll just point it out here: the default rule is to maximize the hit points you get at first level. A barbarian gets 12+con, a wizard gets 4+con. Only at levels after first level do you roll your hit die.

I'm massively opposed to rolling stats, much preferring point buy. And it sometimes strikes me as inconsistent that I do roll for HP. The difference, I suppose, is that HP should even out to an average after enough levels. I might eventually switch over to half-plus-one (alternately phrased as average-rounded-up, though that's a worse phrasing given D&D's Always Round Down rule) for HD rolls, but until then, is there some way to make rolling HD better?

I've played in high-powered campaigns where if you roll below average on your hit die (after 1st), you simply take whatever is on the opposite side of the die. E.g., if you roll a 1 on a d12, you instead take a 12. If you roll a 3 on a d6, you instead take a 4. This obviously had the effect of substantially increasing one's hit points.

It has occurred to me to institute a system whereby, at each level (after 1st), you may choose whether to roll your HD as normal, or take your average HD (rounded up does seem to be the standard, but offering it as an option should probably round down, to make it a choice between the certainty of one result, or a random result that could be much higher or much lower but which will average out to slightly higher than the certain result; it may still be an even choice, because people supposedly tend to pick the devil they know over the devil they don't). But I haven't thought through this idea, and I don't think I'll ever test or implement it.

I've also heard tell of a house rule to this effect: if you don't like your HD roll (after 1st), you can reroll on the next die down (d12, d10, d8, d6, d4). You can continue to do this until you hit the d4.
This has the effect of making a high hit die mean more: not only are you getting a higher number to start with (on average, each HD step increases your HP by 1/level), you're getting more cushion in case you roll poorly. No more barbarian crippled by rolling a 1 for his level 2 HD! Well, he still can, but it's profoundly unlikely (unless my math's wrong (which it is -- see if you can identify why), he has a 1/23,040 chance under this system to wind up with a 1, where before he had a 1/12 chance).
I like this system, and may implement it at some point in the future. Especially because it benefits high-HD classes (i.e., melee) quite a bit and benefits low-HD classes (i.e., spellcasters) not a whit, which is good.