I recently pulled my very old mountain bike out of storage, which wasn't stored properly, and there is something wrong with the tire bead. It makes the tire bump around a bit when I ride. I've tried deflating it and leaving it for a night, but it's still off center. Does anyone have any advice to fix it that doesn't include buying a new tire?

Sure. Deflate it most of the way, mix up a pretty strong solution of dishwashing liquid and water, and paint that onto the bead area with a paintbrush or whatever. Pinch the tire so the sudsy water can get between the bead and the rim. Now begin inflating the tire in stages and see if it'll seat up without too much drama. Any good?

Check to make sure this is actually a bead seating issue and not a casing failure issue.

You should probably take the tires and tubes off the wheel to make sure the tube isn't stuck to the tire. Examine everything while off and remount if it looks good.

If it's a really old bike that has been stored in hot garages for much of it's life and these are the original tires then I'd be looking at the tread to see if the rubber itself is cracking or otherwise separating from the bead or fibers of the carcase. Check the sidewalls for bulges that indicate that the threads in the carcase are separating from a rusty bead wire. Yes it happens on old tires that are badly stored in moist and high heat conditions.

Tires don't go "lumpy" for no good reason. And when they do it's wisest to replace them if remounting doesn't do the trick. It's not possible to fix such things.

If you feel the lump when riding it then the tire is done. If it's just a bit wobbly side to side with the center line tread snaking back and forth that's not abnormal. But if there's an actual up and down lump to the ride that wasn't there before then you're a half step away from total failure and a crash. What's cheaper? Your face or a new tire?