What weaknesses does a lich have? I've found a lot about their abilities and strengths, but not their weaknesses.
In general, they have few specific weaknesses beyond those all undead share. Specific liches can of course have unusual weaknesses.
Since this is for 2e, I recommend looking at Van Richten's Guide to the Lich. It is designed specifically for Ravenloft, but it is generally compatible with almost any of the 2e settings. It goes into background on liches and lays out a number of variations which a DM can use for making custom liches, including unique weaknesses that a particular lich may have.
Of particular note, it suggests that many liches may need to periodically perform elaborate rituals to continue to maintain themselves. Those rituals can be exploited by patient lich hunters because the requirements of the ritual often force the lich to venture forth from their lairs to either acquire substances needed for the ritual itself or to be in the right place at the right time to perform the ritual. In short, they provide an opportunity for adventurers to ambush the lich at a time when the lich is away from its base of power.
The whole process is more about preparation rather than finding a weakness.
A Lich doesn't really have any weaknesses which is part of why they're so terrifying to fight/encounter. If it's something the party stumbled on, they should run, go prepare and then come back. It really comes down to how your party plays it out.
You have to consider what a Lich is: an undead magic user. So it's weaknesses will be related to the defense of it's magic attacks. Silencing and physical attacks are probably best. An issue when fighting a Lich is they sometimes have magic resistance the older they are. In playing, we've had success by using certain magical items (wand of Rock to mud, wand of light for attack/ Rings of protection, spell turning for defense - but remember a ring of spell turning does nothing for area effect spells ie: disintegration). In one fight, we won because a character (out of desperation) grabbed the Lich and the others just pounded it. By grabbing the Lich and pinning it down, it loses it's ability to cast somantic spells, but you still have to worry about anything verbal or material.
Resources: 2nd Edition players handbook, 2nd Edition DMs guide, Red Wizards - #3 in the Harpers series, or more importantly the *Erevis Cale trilogy.
*The Erevis Cale trilogy illustrates the true power of a Lich, but this one in particular was a master of magic and psionics.