Solved!

I thought I would post this for others' benefit - it was a weird one.

Even though I was able to copy the entire music library to another Mac and sync it to the same iPhone with no problem, the more I kept experimenting on the machine with the problem, the more I narrowed it down that it looked like an incompatibility (or corruption) with a file or files. So, I added columns to display the Bit Rate and Sample Rate. Then, by sorting those columns and syncing combinations until it broke, I worked my way through looking for an offender.

It ended up being Sample Rate. For whatever reason, any music file with a Sample Rate over 44.1 kHz (so 48 kHz) would cause the 13001 error (thanks for being so descriptive, by the way, Apple!). So, I just converted the few songs that my daughter had that were 48 kHz down to 44.1 kHz and no more 13001 error syncing music!

I hope this info can help someone some day. It drove me nuts!