Hi Sexycar,
Its good to hear that you are concerned about the up keep of your car. And its understandably so, as the AE86 is a dream car to own for alot of us here and we would also do the same if we were lucky enough to own one.
However, I suspect the reason for the lack of response to your question is because of its vague nature:
1) What do you mean stock LSD? or rather, Does it actually have an LSD? I'm most probably wrong but aren't most stock AE86 come with an open diff? Even if it does , you would need to identify what kind it is to enable us to estimate the cost of the repair/overhaul as a) Multi plate (clutch)= plates needs to be replaced B) Torsen= worm gears to be replaced c) Viscous= can't be repaired
2) What is wrong with it? I'm a firm believer in the saying of " if it ain't broken, don't try to fix it" . Normal fluid change would be good enough in this application . Bewarned that the fluid in the diff should not be discolored ( like your engine oil) which is an indication of a bad diff. The fluid that you are about to change should look like new. Just keep it in the within the recommended interval oil change and use the best synthetic you could find. I would go for a thicker viscosity. like 180 for added " protection"
However, if you still need a budget to work on or if you do find out that you have a bad diff , I would recommend that you go for an aftermarket item, which I was told is rare and expensive; expect to pay between RM1.5-2.5K for a used item