Both Child and I have an external hard drive. The key to both the problem and the solution is that our drives use the same-sized plug, but hers is 9 volts and mine is 12 volts.
Child's drive stopped working on her MacBook. The drive power-light would light up (briefly) when it was turned on, but it wouldn't register with her MacBook. I tried it on my Dell, and Windows would recognize that a device was plugged in, but it wouldn't install it correctly. We took it to SimplyMac, and it wouldn't work on their computers either. Again, the drive would light up, but not register on the computers. I even tried MY external drive (but using her power cord!) on her Mac, but it didn't work either.
The key turned out to be the power cord. The issue was very subtle, because her 9 volt power cord was apparently enough to turn the drive on, but not powerful enough to make it function correctly. Which is strange, because it was the power cord that came with the drive.
However, using my 12 volt power cord, the drive would not only turn on, but function correctly. Maybe this will help someone else...