Not too clear on the AD&D rules with regard to this -- apparently daggers, short swords, and long swords work. How about axes? Two-handed weapons? Blunt weapons? Even ranged weapons?

Strictly in the context of BG2, just about any melee weapon a vanilla thief can use is what you can backstab with.

dagger, club, quarterstaff, long sword, short sword, katana*

*I'm not 100% sure about this one, but my F/T was able to use Celestial Fury with a backstab.

Kuemper's got the right of it. Any melee weapon that a single-classed thief can normally use (without *ahem*). This applies to both games. Going down the list, this includes:

Long sword
Short sword
Katana
Scimitar
Wakizashi
Ninja-To
Dagger
Club
Quarterstaff

However, note that there are exceptions within these categories, but still keeping to the rule mentioned above. A Mage/Thief, for example, cannot backstab with the Staff of the Magi despite being able to use it and it being a quarterstaff, because a single-classed thief could not use it. These exceptions are pure BS, but I have a feeling it's because they coded the limitations to exactly the rule of the single-classed thief. Like maybe being "unusable" for a thief shuts off the backstab; it would be a good example of using one field for two purposes (considered bad coding, but it works).