Low voltage wiring systems

Lower voltage means higher currents, and that can make burn-ups more likely. Possibly some damage has occurred, or a protection device associated with the supply transformer has operated.

Likely the relay is not meant to have power applied for a long time. The smoke you noticed might indicate that the coil overheated and was damaged.

Are there labels on the relays so you can figure out which lights are controlled by each? Are the relays a plugin type? Has the damaged switch been replaced?

Once the switch is replaced and if you can identify the relay and they are plugin remove the suspect relay. Might there be one labeled spare, if so use it to replace the damaged one, otherwise take one that is working and plug it in. If there is any sign of heat yank it out. Else go down and try the switch. If it works, great, now you only have to find a replacement relay. Otherwise, I give up.

Hopefully, the relay has a string of letters and numbers. Input this into a search engine possibly along with the word relay.

If you have any doubt try a Radio Shack store or two. Evan if they do not carry the relay you need they can likely tell you whether you have found a good match or not.

Buy 2 or 3 so you do not have to go through this again. Get some replacement switches also.

Good luck and let me know how it goes. One more question; is your house unique or are some neighbors set up the same way? They would be a good resource for finding an electrition.