Apple Macbook A1181

From the information that you are giving me, I am sorry to say that the A1181 MacBook will not support iOS 10, as the latest iTunes version that came out for it was released before iOS 10 was released, and there will be no more updates for the MacBook. iTunes is what she needs to sync her phone to the computer. What I would suggest is that you buy the A1342 MacBook from either late 2009 or the 2010 model. Both of these computers can be had for under $300-400 if you find the right person or seller online, and they will run the newest Mac OS X (macOS), Sierra. Sierra will be able to run iTunes 12.5 which is supported with iOS 10. Some of the bigger chains that resell mac might not sell these models, so a GOOD seller on eBay might be the best way you can go, some of these sellers might also bundle warranties and other accessories with it.

Another computer I might suggest if you want it to last for a little longer than the MacBook is to get the 2012 MacBook Pro. They have just ended production of that model, so it will most likely still be supported for another few years. they can be found for about $500-1000. They are made out of aluminum and have a better look to them than the 2009 MacBook in todays world.