You Will Need:
-A loved N64
-The offending AV cable
-Q-tips or similar long thin stick-y thing
-Petroleum Jelly (Vaseline or the generic version works fine too)
Step 1: Make Sure the AV Cable Connection Is Actually the Problem
-Is the red power light on? If the answer is yes, move on. If not, make sure the N64's on switch is flipped and that the power cord is actually plugged into a working electrical socket. If that still doesn't work, then the problem with your N64 is bigger than the AV cable and I don't currently have an instructable to help you, sorry!
-Is there no video or sound despite the fact the AV cable is plugged in correctly? If the answer is yes, and you get a "no signal" or similar message on your TV, then you can be pretty sure that the problem is the AV cable connection. If there is sound but no video or vice versa, then it isn't the N64-to-AV cable connection that's wrong and you probably want to check your TV's AV ports or look for a new AV cable completely.
Step 2: Apply the Connector Substance to the AV Port
But there is a very simple way to make the connection all the time, every time! We're going to add a connecting substance that will make sure they always connect. Start by getting a nice swipe of petroleum jelly on your Q-Tip (or other similar device).
Then simply smear the jelly into the AV port on the back of the N64. You don't really have to stick the Q-Tip inside and don't frost it on like you would a cake - just a nice thin smear around the edges will work (when we plug the AV cable in it will take the jelly into the port with it).
Step 3: Apply the Connector Substance to the AV Cable
Step 4: Plug in and Test
I first did this fix to my N64 in May of 2012 and haven't had to reapply the jelly since. I can move the N64 around and knock it without fear of losing the connection. If you're unplugging the AV cable often you might want to reapply the jelly every so often, but only if the connection seems to be getting fuzzy again (you don't want to cake up your ports with the jelly).
As far as I'm aware there's no way the jelly can damage your N64, and if you put too much in or want to get it out, use a hair dryer to heat the jelly up and let it drip out.