87 Supra won't start

[RNEMESiS42]

All right, this all started back last July when my Supra was selectively starting. It'd work fine for a day, then not start at all for a few days, then start working fine again. I got a brand new battery, and that solved the problem completely. Around October, the same problem started again, except now it didn't start at all, no matter what. I replaced the spark plugs first, then the igniter and the ignition coil. Still nothing. The distributor wasn't getting any spark. Since the weather is finally nice again, I thought I'd start working on it some more. I gave it a full oil change last night (it was black and goopy...) and replace the distributor cap and rotor. This morning I charged up the battery fully, and gave it a try. It started barely turning over, and then I gave it a bit of gas, and it caught! I thought that was great. I let it idle for a while, and then drove it around the block, no problems. I parked it back in my garage and turned it off. I suspected it wouldn't turn on again...and I was right. What is going on?!
I'm thinking I need to replace the whole distrutor, maybe? Maybe just the spark cables? I've basically replaced everything that has to do with the ignition, besides the ECU and distributor. I really don't know what else to try (especially since I got it to start up once today!)