You may also find original (used) OEM discs for that machine on eBay as well. When I buy used machines, that is what I do. When I bought a G5 Power Mac a couple of years back I scored the OEM discs for only $20.
The discs for the mini can be had for about $50
»www.ebay.com/itm/Original-Apple- ··· 5wt_1164
While there you can see what that mini is fetching on eBay.
EDIT - I have an OEM set from a now long dead 1.33GHz Mini but they're 10.4.2, not the later 10.4.6.
According to Apple
10.4.2 was the latest actually required. I think this would have been for the 1.33 and 1.5 G4 minis. In later years that 1.2 was replaced with the 1.33 and the 1.42 replaced with the 1.5 although the boxes didn't say it (as least that I saw). The earliest mini's shipped with 10.3.7 with a Tiger drop in disc (ah memories). My 1.42 G4 is still chugging running as a small file server 24-7 going on some 7 years now.
So even if your machine CAME with 10.4.6 you SHOULD be able to boot and install with 10.4.2. Once installed (if it installs) you will have no problem updating it to 10.4.10. That mini will also run Leopard up to 10.5.8 although it can get a little sluggish. There are tweaks to turn off unused features. The big limitation of those 2005ish minis wasn't so much the processor but the 1GB of max RAM. OS X shines with 2GB+ RAM which isn't supported in those 1 RAM slot minis. They also choke running Flash but flash blocking browser plug ins fix that.
Anyway you are welcome to these discs. I'll PM you. If you want to give me a shipping address I'll drop the discs in UPS or Fedex Thursday. They're just reminders of good times at this point.