I voted "average", but I realized that it's actually "player's choice". I just don't have any players who want that choice.
Don't forget that...
Ah, then I will amend my vote.
...you can do this. Click the red "Unvote" in the upper-right corner of the poll, and then you can vote again.
I voted "gentleman's reroll".
My system: You can choose to roll or take the high-average, as listed in the book; however, if you roll, you have a minimum on the die of the low-average.
IOW, if you have a d8 HD, you either take 5 or roll 1d8 with a minimum of 4.
So far, everyone has rolled.
Well of course they would. You're giving them a die that is d{4,4,4,4,5,6,7,8}. Although they have a 50% chance of getting less than the "high-average," as you put it, the overall average is 5.25 for that die compared to the 5 static. They risk only a 20.3% chance of getting less than the static average by level 10, but stand an almost two-thirds (65.08%) chance of being at least 1 HP higher, and they have better chances of getting more than 10% extra HP than they do of getting any amount less than average (32.59% chance to get 55 or more HP). This will make your "low-average min" die statistically superior or equal to the "high-average" static value for all HD used in 5e, as the unused d4 has a "high average" of 3 but the "low-average min" die gives an expected value of 2.75. (The relationship between "high average of die" and your "die with low-average min" is y = 1.25*x-1, where x is the high average and y is the expected value of your die, regardless of what the maximum value is.)
For once, intuitive feel ("wow, the minimum is the normal average? sweet!") and actual statistics match up!
We keep rolling until you roll in the top half of your dice, eg for d10 must roll 6 or higher or roll again.
Wouldn't it be easier to treat lower numbers as their higher equivalent (e.g. add 5 if the value is less than 6 on a d10*)? It's mathematically equivalent (each value from 6 to 10 has a 20% chance of being rolled) and substantially faster.
*Or to use Jester's terminology, "add the low-average if your result is below the high-average." It's the same as simulating a "d3" by rolling 1d6 and treating 4,5,6 as a second set of 1,2,3; it's just reversed.