You’ve gone through the process of updating your OS X to 10.7.2, updating iTunes to 10.5, syncing, downloading iOS 5, and backing up, but after all that clicking and waiting you find that “this device is not eligible for the requested build.”
The culprit would seem to be a now problematic entry in /etc/hosts. Namely, a single line at the end:
Well, this line is not in a pristine hosts file and should be removed or commented out. By removed, I mean delete it. By commented out, I mean modify the line to read only:
Of course, you cannot edit the file in place, so copy it, and perhaps a backup (how about “/etc/hosts.orig”) somewhere to edit, then copy that back to the original place – all as an admin user.
Reboot. Rebackup. Rejoice.