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
Probably not, unless it's just lightly plugged with a thin flash (eggshell thickness or so) you can break through without breaking the overflow channel (either out to the underside or in to the sink bowl.) If it's a plug of thickness similar to the channel walls, I doubt you'll get through it without going wrong in one of those directions.
This should have been inspected and found at the time of original installation when it would more easily have been resolved by return to the supplier for a correct one. If you have a really top-notch supplier that might still be possible.
Best Answer
Take a look at a typical bathtub overflow drain:
That's basically what you'd need. The specific parts depend on your other components, but they're all fairly standard.
Whether there's any point to doing this is completely subjective.