Today I noticed one of the two bolts that mounts the distributor to the front engine cover on my 240z was loose, so I tightened it. The bolt seemed to turn freely, but suddenly just snapped off flush with the front engine cover when I tightened it. I didn't even snug it and somehow it broke off. Now I have to find a way to remove the broken off piece from the engine cover. What's the best way to go about doing this? I realize that I will have to take the distributor off, but will that screw with the timing?

Shouldn't mess with the timing unless you have to remove the front cover to extract the broken bolt :-(. Just be sure to plug the hole with a rag to keep any metal shavings out of the engine.

As d240za2 said, keep a rag over the distributor to keep anything foreign from falling in. Buy a good drill out bolt extractor- Sears has a variety of them usually on sale. Use a drill with a light hand to keep the speed low and use some machine oil as well.

Good luck!