this is an old question but hopefully my answer will help some others.
if the system is known to work then you either have an issue with the game's contact pins or with the capacitor on the cartridge board. If you have time and a few tools you can figure it out. Unfortunately you only have one game. You really should grab another so you can rule out the system as the issue.
many people say to clean the game's contact pins you should use rubbing alcohol. I find that using metal/aluminum polish works extremely well. I use the Mother's brand in the small round can. You should only need a good cleaning with the polish one time and the game should not give you issues ever again. It works wonders on NES carts. To apply it use two pieces of old t-shirt. One to apply the polish and work it in until it turns black. The other to clean up. Keep the polish off of the chips. Use masking tape if you are unsure. You will be amazed how nasty brown pins slowly turn black then come out blinging after you clean them off.
The other potential issue is specific to n64 carts. Some have a capacitor surface mounted on the board. Over time capacitors will fail, as evidenced by what happens to old sega game gear units. You can open the cartridge and get the numbers off of the side of the capacitor and radio shack usually stocks a replacement. It's a quick replacement if you know how to work a soldering iron.
edit: since posting this i've found out about boot chips also being an issue for a black screen so check those too.