Thanks for all the advice!
I also am hesitant to try restoring it on my own. But I live in a small town, not near any cities, and there's no where I can bring it to ask what I should do.
Since one of you said gazelle hide, and after looking at it a bit more, I'm starting to lean towards that. There's a small band of hair around the very edge of the drum that I hadn't looked at very well. But now that I do, it does not look like zebra hair. It's all just dark brown. So I think gazelle is probably more accurate.