I finally got down there yesterday and removed the sucker. I bought a new one, seeing that I'm not the best solderer in the world. Everything was going great until....I needed to get the damn thing back in. I don't know who designed the undercarriage to the 90 Civic dashboard, but he must've had REALLY small hands. To get that 10mm bolt back into place, with it's tight threading, you've got to take the entire lower left plastic facing off. If not, you'd better have tiny fingers and steady hands. I didn't have the time to take it all off, so I used a pinch clip to hold down the metal arm of the relay. So far, so good. I drove the car in 90 degree heat for 5 minutes at highway speed, pulled back into my carport, turned it off, and turned it back on! It'll also now start after sitting in the hot sun.
Upon inspection, I noticed quite a few possible dry joints in my old relay. Perhaps that's why the problem has become so recurring as of late.