I want to clean inside the hole of the bathroom sink that is used to control overflow. I can stick my finger in there, and rub off a small circle's worth of stuff, but I'd like to clean it more than that. Any ideas?
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
I tried using spray cleaner, vinegar and baking soda, etc. but nothing worked until I ordered this brush: https://www.amazon.com/gp/product/B072B9ZX9K
The gunk that it pulled out was quite impressive. The ability of the overflow to grow biological goo is the reason that the Int'l Ass'n of Plumbing and Mechanical Officials (IAPMO) did away with the requirement for sinks to be manufactured with overflows a few years ago.