Can't restore music to iPhone from iTunes
Note: This is all done on a Mac. Just look up the PC equivalent of the following steps if required.

How to recover past information:

Option 1 - Restore iPhone 4 directly from old computer:


Connect iPhone 4 to old computer, open iTunes, put the iPhone in DFU mode, (while holding alt/option key) select Restore after iTunes has detected the iPhone 4 in recovery mode, and select the most recent backup of that iPhone 4.


Option 2 - Copy iPhone 4 backup from old computer, and place on new computer to recover:


On the old computer, go to ~/Library/Application Support/MobileSync/Backup, and copy the most recent backup of the iPhone 4 and transfer it to the new computer.
Connect iPhone 4 to old computer, open iTunes, put the iPhone in DFU mode, (while holding alt/option key) select Restore after iTunes has detected the iPhone 4 in recovery mode, and select the most recent backup of that iPhone 4.


How to create the ability to sync all purchased/future purchases of music and apps:


Open iTunes.
Sign out any pre-existing AppleID/iTunes Account once iTunes loads.
Sign into desired account of individual (your relative's I'm assuming).
In iTunes, select Store -> Authorize This Computer...
Rejoice!
Have your relative use iCloud for backup to make life easier in the future.


Let me know if you have any more questions, or need any futher explanation. Enjoy!