IMac – Why is the Time Capsule backup getting stuck on “preparing backup” or “cleaning old backups?”

You don't need OS X Server for Time Machine backups. Regular OS X works fine so he could just repurpose the white MacBook into a backup server with the external hard drive and get her an Air instead.

I agree that he should confirm that he doesn't have network issues with both his wireless LAN and WAN. Losing the Internet connection is more likely to be a modem or provider issue than a TC issue. Switching the connection to another router can confirm it. The new Apple AE is a good choice but an Extreme might be better if he needs multiple Ethernet ports at the router location.

This isn't normal, but you have two options:

1. Just wait for all the processing to finish or error out.
2. Help the process along.

In this case, an initial backup can take a long time, but the way to help the process along is to delete the .../whatever.inProgress file. You could also try backing up to a new drive to make sure it's not a problem with the Mac as opposed to a problem with the backup drive or just a lot of changes to reconcile.

See this other answer for details.

https://apple.stackexchange.com/a/65801/5472

In cases where you run into this repeatedly, you can either try to isolate what folder takes so long to back up. On a first backup, I've tried to exclude the Users folder (or several folders at the base of the boot volume if they have more than 10 GB of data) to make sure that the filseystem is OK and the backup drive is OK. At that point, you will see backups taking a minute or so for each incremental backup. You can then relax the exclusion list and know if it freezes, then you have a problem in a specific folder.

The other way to attack this is run disk utility (or another tool) to repair the directory structure or make a back up using another tool and wipe the drives / restore them from backup.