Having thought about this before...
Base Systems
1: Crouch and prone positions (instead ranged accuracy, lower enemy ranged accuracy, better sneak and better use of cover at the cost of higher move cost, worse melee and the AP cost of position shifts). Full control of companions in combat. Guns take accuracy penalty based on distance to enemy (handguns work fine even when adjacent, carbines work fine unless the enemy is directly adjacent, long rifles take large penalties without multiple hexes clear, and big guns need hexes all around instead of just between). Basic tactical layer things.
2: All humanoids have a base AP of 5 and gain another 3 AP per point of Agility. This frees the system of the retardation with breakpoints that come with a single digit AP count (a firing cost of 4 means a 8 AP character you can fire twice as fast as a character with 7 could but a character with 9, 10, or 11 fires exactly as fast as one with 8) and allows including minor actions with AP cost. Also allows "damage" to AP without being instant win.
3: Suppression fire is a thing. Misses (so long as it was reasonably close) will damage the AP of most targets (robots and some "feral" mutants which are so rabid to have no concept of morale are immune) on their next turn depending on "noise" of a cartridge. This means burst isn't just a quasi-melee ability, but an effective tool to shut down certain dangerous enemies. Some weapons/cartridges have pen modifiers making them high damage low pen (black powder rifle or 12 gauge slug) or high pen despite lower damage (5.56). Suppression effect is resisted by Charisma so that has a combat use.
4: Melee users get a free attack on anyone leaving their reach unless the mover pays extra AP. (makes it harder to run away from melee)
5: Rather than opening the inventory costing AP, using something from it or swapping weapons has a per item AP cost, and a character has a limited number of "ready" slots limited to smaller items (grenades, pistols, knives, knuckles, healing items, lights) where this cost is reduced. Less shenanigans, more use for smaller items.
6: Light is important to hit anything, but portable electric lights are common and their power isn't tracked short of being able to remove the battery (primarily to fuel energy weapons). Lights can be worn, thrown or placed (and setting something on fire creates light) so you have to plan lighting to make enemies vulnerable without making yourself vulnerable. Later game lets you find night vision, but you get penalties in heavy light and a penalty to aimed shots when using it.
7: Big guns are not a thing, just a pairing of strength and guns.
8: Projectiles are simulated rather than be simple hit chance.
Health:
1: HP is ((Endurance*10)+(endurance*2*(level-1))). This means HP depends more on Endurance, and is large enough to allow for weak attacks to deal damage consistently (instead of quickly overwhelming HP or having to be rounded down to zero) while small enough to be appreciable (at level 20 a max endurance character has only 400 HP compared to his starting value of 100, and a 5 endurance character has a starting value of 50 and 240 at level 20)
2: Single shots from weapons are damaging (a rifle or heavy stab with a spear from an average user might deal damage in the 30s). Later weapons are more dangerous because they can be used faster and are accurate enough to hit more vulnerable areas. HP sponges are never fun.
3: Bleeding is a thing for edged/pointed weapons and guns (not blunt or energy), dealing damage until the character receives healing.
4: Characters under half and under a quarter of their max HP take penalties until healed (by any amount). Combined with the above, it stops "spreading damage is useless because 1 HP enemy is as deadly as full HP enemy" nonsense.
5: Use a DT system for armor and let it nullify damage entirely, but armor only protects limited parts of the body (typically just torso for body armor and head for helmet). This means aimed shots have a point to exist beyond eyes for damage, groin in melee for stun and torso because you have nothing better to do with that one AP and might as well get a slightly better crit chance since the enemy can't reach you so the extra AC doesn't mater. Now the arm and (especially) legs shots are how you damage armored targets you can't penetrate without the huge accuracy sacrifice of a face shot. It also makes power armor scarier and more distinct: It's not just tough, but it protects the whole body.
Weapons
1: Use the rules Sawyer made for NV when designing weapon stats. No "strictly better" upgrades, all weapons should have a reason to use them over the rest of their tier, a reason to use the rest of the tier over them, and not all weapon types ("shotgun", "pistol", "distance rifle") have an entry each tier.
2: Guns are reasonably common, but most ammo, especially exotic ammo, is relatively rare and expensive due to being a limited resource and rarely stored in a condition to last the end of the world. Black powder, 22lr, and shotgun shells are reasonably common (and more than enough for a gun specialist to never be totally out of ammo, so long as he's willing to use an inferior weapon) due to post-war production and sheer pre-war quantity respectively, but that fancy .44 magnum is just a paperweight if you expend the cartridges you found it with and can't get more. Needing ammo is Gun's weakness compared to unarmed, melee so it should actually mater.
3: New Vegas's cartridge reloading system is in. Turning stuff that's useful for other characters (instead of just useless except for crafting) into stuff useful for you and turning limited resources into your choice of consumables remains one of the few crafting systems that's actually fun. Also makes explosives more viable as primary if you can make improvised grenades from powder, lead and scrap metal.
4: Ala Morrowind, fodder enemies use crap weapons with minimum trade value for their weight (post-war muzzleloaders, single/double shotguns, plinking rifles) where their limited ammo supplies are worth more than the guns. They're threats because their numbers make up for their low fire rate/damage and give them more chances to "get lucky". This cuts down on bandits being gold mines and picking up everything to loot (some shit just isn't worth it), but the changes to HP and AP means their numbers will keep them a threat.
5: Manually operated guns cost the same AP as a self loader to fire, but need the pump/rack done after firing (pump action, bolt action) or cocking done before firing (single action revolvers, flintlocks) as separate actions that cost AP. This allows for more system finesse than a weapon just being slower to fire and forces more tactical options (do you wait till you're in cover to work the bolt or do you do it immediately? Ambushes are more effective.). A SA/DA handgun can fire DA instead of cocking at a cost of lower accuracy.
6: Energy weapons are relatively common at all stages, instead of non-existent early, rare mid and common late like F1/F2 (which is just unworkable without meta knowledge and broken with it). They have low suppression but generally low strength requirements and tend to have secondary effects (for example lasers have high armor AP, plasma sets targets on fire, electric damages action points if it hits, concussion pushes enemy back/down, wave has low damage but ignores armor and thinner cover entirely) which make swapping between multiple energy weapons viable. The early ones are "recharger" systems, which automatically recover ammo each round (but less than you could fire), or take standard "domestic" batteries. Some have internal batteries and can take any type of battery but need hours to pull that charge (making them quasi per-day expendables).
7: Smoke grenades are a thing.
