The diagnsotic is a basic terminal window that is used to connect to Tivo as it boots. It's most useful function is that it can be used to switch the boot partitions from the current setting to the backup partition if something gets hosed. This is done from the factory settings menu.
If you have a serial cable, start up your telnet client and cycle power to the tivo. As soon as it gets power(you might want to have one hand on the switch, one on the enter key....or get a friend to do one) toggle enter. You should get a password prompt...hopefully it's the default "factory". If not search on the TivoCommunity forum for how to reset it. Once your in just go to eXtended menu, boot Parameters, and change it from root=whatever to the one you want. Hit b to boot and you're good to go.
The standard serial params are 9600,N,8,1.
They can be kicked-up a bit while doing transfers, but the default boot setting is as listed above.
Typical partition layout of tivo is listed below:
1 partition table
2 boot
3 kernel
4 root
5 boot
6 kernel
7 root
8 swap
9 /var
10 mfs master partition
11 mfs media partition
varies from here depending on upgrades and version
Note there are two sets of boot, kernel, and root. One is active (pointed to by the root=/dev/hda4 or root=/dev/hda7 command in your boot paramters. The other is not normally used. They copy the new kernel and build a new root partition for it on the set you are not using at the time (that one is mounted as read only). Then they set up the system to swap over on next reboot and schedule a reboot for some time like 2am when they don't expect you to be recording or watching tv. If all goes well, you wake up and have new software. If you have installed goodies like tivo web, then you might just loose part of your hard work. If you changed something about how the system works that would affect the upgrade process, it can hang.
In any case, at least backup your hack directory before you go much farther. rc.sysinit can automatically destroy and rebuild the /var partition if it runs low on space. Putting hacks there is convienient, but not necessarily permanent. Creating your own hack partition is safer, but links to it will be blown away on an upgrade. The partition will not usually be blown away though.