How do I clean a sink overflow drain?
First question: is the overflow blocked? If you fill up the sink and leave the water running, will it spill onto the floor?

If no, don't worry about it. There's may be some mold or soap scum in the drain. If that wigs you out, a bottle brush and some bleach should do the job.

Second question: is the drain blocked because your child pushed bits of toilet paper into it? If yes, then you'll have a solid mass of papier mache in the drain, and the only solution will be accessing it from the bottom (which means removing the tailpiece). My suggestion below won't help.

So, assuming that you have a "normal" clog in the drain, such as what might happen if you shave with a full basin and hair floats into the overflow, you can get a plastic mini-snake that is designed to pull hair out of a drain.

Here's one example that Home Depot sells: http://www.homedepot.com/buy/plumbing/drain-openers/brasscraft-zip-it-bath-and-sink-hair-snare-bc00400.html