For a very long time (actually, since point-buy ability scores became standard), my biggest bugbear with D&D and D&D-like systems (PF, SWSE, and similar) has been rolling hit points. The major reason for this is that for the entire 3e-era I only ever rolled a '1' for hit points. (I should note two key caveats to that: firstly, and most importantly, that was actually very few characters over very few levels; secondly, it's a streak that ran from the release date of 3e to the release date of 4e, and which has finally been broken in a 3e game.)
It can be fun playing a character with minimal hit points and trying desperately to keep him alive despite a critical weakness. But it's absolutely not fun after the first such attempt. Consequently, hit point rolls became the one and only roll one which I would cheat - if the DM absolutely insisted on random rolls and didn't institute some sort of a minimum and/or reroll (which was extremely unusual), then if the dice came up a '1' that would quickly be... massaged.
It was always interesting just how angry that made some posters on ENWorld. :)
As with ability score assignments, I went through a whole bunch of house rules for hit point generation in 3e, before finally settling on one that I liked. One of the issues I had was that the "max hp at first level" interacted strangely with the multiclass rules, so that a Rogue 1/Wizard 1 was rather better off than a Wizard 1/Rogue 1. (This applied to the skill point assignments as well, which is why that's my go-to example.) So I had a table with some correcting logic that was just really confusing and served mostly to make a teeny tiny adjustment just to fix the issue.
In the end, I settled on fixed hit point assignments based on class: classes with a Poor BAB got 3 per level, classes with Medium BAB got 5, and classes with Good BAB got 7. (Which meant that the Ranger effectively "moved up" to the same as Fighters, while Barbarians "moved down". That was a feature, not a bug.) In particular, though, the "max hp at first level" was replaced by a flat +5 bonus for all characters. So a 1st level Wizard got 8 (3+5) hit points, rather than the standard 4, while a 1st level Fighter got 12 (7+5). (Again, a feature, not a bug, as was the fact that characters thus had significantly more hit points than the average of the dice roll.)
Anyway, 5e...
The stated rule in 5e is that at each level the player gets a choice: either take the average (rounded up) for the class, or roll your hit points. Per the rules as written, there is no provision for rerolling or an imposed minimum on the roll, or anything like that - if you roll, you have to live with the consequences.
All of which is absolutely fine by me. Faced with that, I would always choose to take the average so I'd be happy. For those players who really want to roll, well, the option is there too. And over the course of a campaign it will come out more or less the same anyway (you're slightly better off taking the fixed value, but only by a tiny margin).
So I'll be going with that. I do have to note that of course I expect players to play their characters in good faith. So if you choose to roll, be it hit points or ability scores, you're expected to play the resulting character without undue complaint, without cheating, and without busily 'suiciding' the character. (But, these days, I really don't expect that to be an issue. It might have been with past groups, but not now.)
That said, I do have a couple of variants I've been bouncing around for a while, which I think it may be worth adding here...
Reroll Per Level
I forget where I first saw this one, but I'm pretty sure it's not something I came up with. But here it is:
Whenever you gain a level, reroll your entire hit dice. If the result is equal to or less than your current hit point maximum, you instead gain 1 hit point. If the result is greater than your current hit point maximum, you instead take the new total.
For example, suppose a 3rd level Fighter (no Con bonus) with 21 hit points reaches 4th level. At this point the player rolls 4d10 for hit points for the new level. If the result is 21 or less, his hit point total instead increases to 22. If, instead, he rolls 31 then his new total increases to 31.
The effect of this is to 'smooth' the hit point progression over time. If you happen to roll badly at one level, then it's much more likely that your score at the next level will be higher, and so boost your total. Conversely, if you roll particularly well at one level, it's much more likely your next level (or two) will see much smaller gains. Over the course of the campaign, characters should tend towards the average for their class, but do so fairly organically.
Reroll Per Long Rest
I'm pretty sure this one is mine, and it's also specific to 5e.
Characters start an adventure with the maximum possible hit points for their class. (So a 3rd level Fighter starts with 30 hit points, plus any Con bonuses.)
Each time they take a long rest (and they have taken at least 1hp damage), reroll the entire pool of hit dice. If this is less than or equal to their current hit points, then they remain at their current hit points but their maximum is reduced to their current level. If the result is equal to or higher than their maximum hit points, their maximum is reduced by 1, and their current hit points rises to this new maximum. If the result is between their current and maximum hit points, then their maximum is reduced to the total rolled, and their current rises to this new maximum.
For example, if our 3rd level Fighter had 30 hit points but has taken damage to reduce him to 15. He takes a long rest, and so rerolls his entire hit dice. If the result is 15 or less, then he remains at 15 hit points, but his maximum is likewise reduced to 15. If, instead, the result was 30 then his maximum would be decreased to 29, but his current hit points would increase to 29. Finally, if the result was 22 then both his maximum and current hit points would change to 22.
Obviously, when the characters take a "between adventures" rest, their maximum hp totals go back to, well, maximum. (In game, that's probably something like "take a week off".)
The effect of this is that characters will gradually become more and more beaten up as the adventure progresses - they will start with a very high maximum hit points, but this will gradually reduce as their injuries take their toll. I think it's actually quite an elegant solution to the "long rest heals everything" issue in 5e - yes it does, but it also redefines that 'everything'. It also has the consequence of discouraging injured characters from casually taking a long rest (and likely reducing their maximum total). Though it does encourage a gamist artifact where the group really should exhaust their available healing before taking that long rest (to mitigate the loss). I'm not sure that's really an issue, especially stacked against "long rest heals everything".
Having said that, it's also sufficiently complex that I'm not going to bother using it!