Going WAY back to the basic set, as a DM I threw a gaggle of ropers at the PCs in a small cave and watched them crumble. The roper back then did 1 STR damage with each of it's tendrils. The two fighters got overwealmed and their damage stats dropped, the cleric couldn't do a thing about it, and the magic user wasn't dealing out enough damage with the limited spells in the small cave(no fireball, lightning bolt). The rogue was the only one who out ran them, and was the savior for the party by later finding the roper lair and dispatching them in their sleep.
Thing was, they shoulda been able to handle the ropers, but the PCs had no idea what they were and charged ahead only to be tagged out helpless. If some PC or NPC had any type of weapon like that, it could easily get out of control.
There are spells that enhance/detract from abilities as well, and using a bunch of these would produce the same effect, so I see no reason not to have it.
It's a creative idea though, and I wouldn't disallow it entirely, just make it very hard to come by and with more limited powers. I would have any weapon do a max of 1 STR per blow (with FORT save to negate) in addition to normal damage. I would suggest the temporary damage heal at 3 points per hour + any CON bonus.
The advantage to having any weapon to do any type of ability damage is clear, as weakening a dragon would be the better route instead of clobbering it with massive amounts of damage. But by doing only 1 STR per strike, it would take a while to wear opponents down, and having the save to negate would balance the game, as not many tough dragons fail their saves 26 times in a row. It also balances out with low strength opponents as well, as the damage may be better to deal than the STR reduction.
I also like the thoughts about subdual. Nasty opponents could be weakened by something other than a weapon, such as an enchanted pillow, dealing no damage to an elderly wizard that needed to be captured without harm...although grappling would work just as well.
Now we need to debate about INT reducing arrows, DEX removing flails, etc.