“How do you test a relay to learn why it went bad?“
First, if you have an ohmmeter and a spare, working, similar relay, measure the resistance across the coil contacts. There should be a diagram of the internals of the relay on the case somewhere that will show you which contacts to test. If the resistance is the same as the spare relay, the coil is probably OK. If the ohmmeter shows infinity on the dead one and some measurable resistance on the spare, then somehow you have overloaded the coil and you need to look at reasons for that, like a short on the switch side, but that would seem rare to me. Usually a relay coil only succumbs to over-voltage or if it is a short duty cycle design that is constantly on.
If the coil tests out OK (same as the spare) then remove the case from the relay and look at the make-and-break contacts at the end of the long metal wiper. If I’m right, they will either be (A. ) welded together, (B.) burnt and pitted (making lots of resistance and no longer working) or (C.) totally gone since they were burnt into oblivion.
I would go with B above. That means that whatever current they were switching on that side of the relay (it could be a double throw, switching between two circuits/lights) is drawing more current than the relay contacts can handle. That could happen if the headlight circuit shorted to ground between the headliht and the relay.
If they shorted, they might also weld the contacts together and that would indicate a short to ground somewhere on the light side of the relay, overcoming the rating of the contacts.
Regardless, the power side of the relay should have a fuse between the relay and the power source to protect both the relay and the headlights, so battery to fuse to relay to headlight, NOT battery to relay to fuse to headlight. Got it?
Happy hunting.