1880s sounds right, thanks to your research. The music box was probably made in Switzerland, where most of these were made. The case looks like rosewood with satinwood inlay and some ebonized (NOT ebony, but ebonized wood) trim. I don't know what kind of impact that broken drum head will have on value, but if it can be fixed, do it. If that were repaired in a historically sensitive manner, this would probably be worth $500-$1K, depending on how the finish on the case cleans up and how good it sounds overall.
Nice piece! If you do a little restoration it could remain as a beautiful and functional family heirloom for another 125 years!