How old is the Mac Mini? Migration Assistant doesn't support moving from 10.5 to 10.10 as far as I can tell.
You have a few options, but the HD option seems like a good idea to me because you can make a full bootable copy of the ageing hardware & keep it as an independent backup. The G5 may not boot from USB, so you may need a USB + Firewire drive.
If you want to copy via the network you will need to enable file sharing on one of the Macs & then use Finder to connect to it.
Another option is to use Target disk mode…
OS X Mountain Lion: Transfer files between two computers using target disk mode
You will need a Firewire to Thunderbolt cable (or just Firewire if the Mini has that).