IPhone – How to backup all the jailbroken iPhone apps saves before restoring to new firmware

1. Make sure you have appsync and installous from cydia installed on your iPhone, installed from the source
2. The app has to be a .ipa file. Place the file in /private/var/mobile/documents/installous/downloads.
3. Open up installous and install from there.

Or use iPhone Explorer as described in this LifeHacker post:

Using iPhone Explorer, you can backup or copy apps (and their settings) to your computer or to other iOS device—and it can do all of that without a jailbreak. To backup system files or default applications, like iOS's built-in calendar, you'll have to jailbreak your iOS device. (Even if you're not interested in backing up apps, iPhone Explorer is still great for mounting your iOS device as a disk.)

The last thing you can try to do before sending it to repair is making a new install by Erasing All Content and Settings. There are many ways to accomplish this and many layers of wiping. One of them is through installing a new firmware (ipsw) file just in case the one iTunes download is somehow corrupted.

But doing anything beyond the regular Apple way is entirely up to you as the process are even more time consuming and not all that simple.