Another idea I just had is this for when you want to roll:
If you roll below half, you must keep the roll. However, on your next level, your minimum is your Hit Die minus your last roll.
For example: you have d10 Hit Die.
At 1st level, you begin with 10 hit points.
At 2nd level, you roll and get just a 2!
At 3rd level, your minimum hit points for your roll is 8 (10 HD -2 prior roll). If you roll below 8, you get 8.
This way if you roll badly for a level, you have to accept that and "suffer" with it for a level. However, when your next level comes, you survived the bad roll and are rewarded with a minimum to make up for it.
I think the main problem with rolling comes that you might, due to random chance, be unable to perform your role in a party. I have a a Wizard. They've come very close to death on a few occasions, but in a lot of battles, their hit points don't matter because I play very cautiously, and don't often expose myself to danger (which shocked everyone in our last big battle where I ran out and Vampiric Touched a baddie to death!).
I've noticed (and had people argue about it at length) that it's far easier to function as a ranged character in 5e (yes, even if you don't have melee types, you can build a character who will be very hard to catch in melee) since most of the woes melee characters have to deal with don't apply to ranged combatants. There are many enemies you don't want to get close to. Some enemies you can't get close to. But I digress.
On the other hand, if you play a character who is built to be in melee and take a good share of hits, a low hp total can make it almost unplayable. Cautious and "in melee" don't really mix, as most characters have no way to avoid taking damage other than wearing the most armor possible, which most people do already as a matter of course.
And even if you do have fantastical defenses, you can still be taken out by a huge monster crit (which totally exist, not everything relies on multiattack to get damage in. One of my favorite examples is a giant ape tossing a rock at you for 14d6+4!). Many monsters/NPC's get bonus damage dice whether or not it's justified (looking at you, Gladiator).
Imagine a world where we have a slippery eel Rogue who uses a shortbow and cunning action Withdraw to avoid enemies who has 85% of maximum hit points, and a plate and shield wearing Fighter with 40% of their maximum. In one case, the already cautiously built and played character may seem invincible, especially once he gets evasion and uncanny dodge.
The Fighter who has basically one ability to mitigate damage? Not so much maybe.