No more q-tips! No more alcohol! No more blowing in your cart! Did you know alcohol actually damages the carts? This is a much more effective, safe and long term way to restore the connection.
To do this, you need to open the game cart up, which takes a special bit. Search your favorite shopping/auction site for "NES security bit" and you'll find exactly what you need. To scrub, I used a 3M Heavy-Duty Scouring Pad (a.k.a. Scotch Brite), available at any store that sells cleaning supplies.
So, the process is like this:
1) Open your game up
2) Take the board out
3) Take out your scouring pad and get it wet
4) (GENTLY) scrub the connections clean, make sure they get 100% shiny! Be sure to be gentle, you don't want to scrub off any metal, just that top layer of gunk (the video makes it sound more abrasive than it is, you don't actually even hear the scrubbing in real life).
5) Rinse off the connectors to wash off any remaining grime or scouring pad debris
6) DRY IMMEDIATELY
7) Reassemble cart
8) Pop the game in your NES and watch it boot up FIRST TIME!