Thanks PoD, that was helpful!
The relay is held together with a ring of glue like substance that can be removed with a small screwdriver. Once removed, there are four long tabs that hold the cover on. A gentle pry with the screwdriver will allow the cover to pop off.
Here's what I found tonight after I tore into it. The metal connectors are held in place inside white plastic parts which in turn, are held in place by a series of black plastic "paws". At some point in the car's history, someone had shoved the connectors in when they were not aligned properly and it popped the circuit board loose. This had the effect of the connectors not quite reaching and breaking contact now and then.
Once I snapped everything back into place, I used a small tie wrap to secure the parts.
Now all I have to do is seal the parts back together and it's done!