I cleaned it up and wrapped it up with electric tape and a few days later my girlfriend smelled the burning again... checked it out and there's a hole fried through the plastic connector. The part that goes to the ballast is fine said:
I agree with the earlier poster who ID'ed it as an ARO 440 ballast. Those are basically icecap knockoffs.
What's probably wrong is that one of the pins isn't making proper contact with the wire in the harness. So when the light turns on, it arcs to complete the circuit, burning your electrical tape and harness. That or salt creep in the harness is causing it to burn the tape. I wouldn't run something that could cause a fire.
I just had an ARO that looks exactly like yours knock off a few weeks ago. I replaced it with a 660, with no issues so far. The hardest part was wiring the new harness, icecap makes you wire the power plug and ground wires.