I assume you are connecting this to a modern (LCD/Plasma) TV. If so, many of those TV's don't like the composite signal from older hardware like the TI. The fact you are getting a black screen (instead of the no-signal blue screen) tells me "something" is being sent to the TV and it's attempting to lock on.
Without any testing equipment or a spare, confirmed working TI, to test, it's hard to determine the issue.
So a few suggestions would be:
1) Try to locate the original RF modulator and skip composite. You may have to find an older (preferably CRT) TV to test it on. The picture will be fuzzy but should be legible. This will tell you if your composite cable is good and if the TI is OK.
2) Try to locate an older TV. Especially a CRT TV that has composite inputs. This will rule out your modern TV.
3) Finally, it might be worth sending it in to someone on the forums for some troubleshooting. I'm sure there are people in here that would be glad to help. I've had some hardware experience so I might be able to do it. But it would take me some time as I am currently moving my office downstairs and things are a mess right now.
4) Buy a working TI off eBay or CL. The good news is that TI's are still affordable and haven't skyrocketed in price like Apple's and Commodores.
Keep us informed. It's awesome that you are trying to revive a TI for your boyfriend.