To me, there are actually a number of issues under the broad heading of hit points, and a few "dials" which could be tweaked, depending on what gaming experience you want:
1. Hits to Kill: At the most basic, hit points are simply a more granular way of working out how many successful attack rolls (hits) it takes to kill a PC or a monster. Lower hits to kill (lower hit points compared to "standard" damage per hit, however that is defined) would make characters and monsters more fragile. This would (of course) result in shorter fights, but also (perhaps more importantly where preferences are concerned) swingier fights - in the sense that luck plays more of a role in determining the final outcome. Higher hits to kill would result in longer fights, but give player decision-making a larger role in determining the outcome of the fight. Possibly, the game could have some dial that allows players to set the "hits to kill" level wherever they like: those that prefer shorter, swingier fights could set it at one or two, and those who prefer longer fights that place more emphasis on player decision-making could set it at three or four.
As a side note, you could cut out hit points entirely, and run the game completely on a "hits to kill" basis, but the increased granularity of hit points allows for more marginal trade-offs, e.g. 10% higher accuracy for 10% lower damage (or vice-versa), 10% lower defenses for 10% higher damage (or vice-versa), etc. If there was no middle ground between dealing one "hit" worth of damage and two "hits" worth of damage with a single attack, the two "hit" attack would probably need to be balanced somehow, e.g. with rarity or with significant penalties to accuracy or defence.
2. Hit Point Pools: The simplest approach, the one that most people are used to even if they do not play D&D itself, is to just have a single pool of hit points: any damage taken is deducted from this pool, and any healing received is added to this pool. At higher levels of complexity, you could differentiate bewteen physical and non-physical hit points (a VP/WP system) or even come up with various other hit point pools such as divine favor or luck, ablative magical protection, etc. Each of these pools could have different and distict mechanics for depletion and recovery. The advantage of a single pool is ease of tracking. The advantage of more pools is they can add to immersion and "realism", especially if each pool has a distinct recovery mechanism (physical wounds can be mitigated with mundane skill but require time or magic to heal completely, vigor can be restored with a short rest, divine favor requires you to act in ways that please your deity, etc.).
3. Wounds and Longer-Term Effects: Again, the simplest approach that requires the least amount of tracking would be to have none, and assume that the characters never sustain long-term injuries. A slightly more complex approach would be to impose minor penalties that can be removed after an extended rest. An even more complex approach would be to have a disease track-like mechanism which would allow injuries to worsen or get better over a number of extended rests.
4. Ease and Efficiency of Healing: This might be one of the more complex to adjust. Of course, characters with access to frequent, significant healing would be more resilient (and powerful) than characters without. However, some players prefer games that feature hit point attrition more significantly. One possibility could be to provide a varity of options for healer-type characters: the 4e standard per encounter healing, daily healing abilities, but balanced either in terms of frequency (usable more times per day than encounter-based healing can be used per encounter) or power (each use is more effective than the encounter-based healing ability), or gaining other abilitities to compensate for fewer or less effective healing.
