The trace option is probably a good place to start in finding your culprit (my preferred method) - although i'd probably create a custom trace in profiler say tracking only items that had a lengthy duration or high logical reads...

I also wanted to ask: Does this server have Reporting Services installed? If so, you may want to check and see if someone has scheduled a report to run during the night. If so, depending on the query/procedure used in the report it might be the cause of the sudden spike in TempDB

Just a shot in the dark, but may be worth a look