NOTE: 3.5's Unearthed Arcana introduced a similar system to the one outlined below. I have never played 3.0/3.5/4.0, so was unfamiliar with UA's integration of this concept into D&D when conceptualizing and writing the post below, and only found it in retrospect as a result of research for this post. I also believe that the overall mechanics of later D&D editions are substantial enough that the system below would be more in line with OD&D than almost any other edition.
Here's where my D&D thinking is going right now...
A monster's (or character's) hit points are treated as their stun rating, and their hit dice are treated as their body rating; the stun rating would include hit points modifiers to their hit dice (e.g., the "+2" in "3+2" would be included), but the "body" rating would ignore it (e.g., a 3+2 creature would only have 3 body points). Character hp/HD would work the same.
– a small weapon (e.g., a dagger, an arrow, etc.) would do 1d6
– a medium weapon (mace, sword, axe, etc.) would do 2d6
– a 2-handed/pole weapon (halberd, 2-handed sword, etc.) would do 3d6
Reducing a monster/creature/character to 0 hp (or lower) is simply knocking them unconscious. But reducing their HD to 0 or below would kill them. The number of rounds a creature remains knocked out/unconscious is 1 round per hit point below 0 (e.g., -3 hit points would mean the creature remains unconscious for 3 rounds). Kill shots are automatic for anyone attacking a knocked out creature (basically the same as if they were put to sleep).
In this context, a 5th level MU and a 5th level fighter would take the same number of body points to kill them (after all, they are both rather experienced by this point), but the MU would be knocked out much sooner than the fighter (assuming, of course, an median number of hp rolled on variable HD).
Now, imagine an encounter where a party of first level characters is locked a battle with a bunch of 1HD creatures (e.g., skeletons). The MU is out of spells for the day, so he pulls out his dagger; he's got a decent chance of knocking the skeleton down/out for a bit, but only a 1-in-6 chance of killing it on any successful strike. By comparison, the fighter pulls out his halberd; each time that fighter lands a successful hit on one of those skeletons, he's got a 3x greater chance of killing it.
Healing would have to be re-figured overall. For example, a healing spell would now need to restore a limited number of body points (rather than stun/hit points). And since hit points restore more quickly, healing would be more directly related to body points.
I don't see reduced HD ("body") points affecting attacks. E.g., a 5HD creature would not attack/save as a 4HD creature if reduced from 5 to 4 body points. Experience/ability doesn't go away as these body points are lost.