Does the Mini have an Intel CoreSolo. CoreDuo. Core2Duo...?
Early models of these shipped with Tiger OS X 10.4, too.
If the computer is an intel-based Mini, and not a Power-PC based Mini
then it could run an OS X later than Leopard 10.5.8. The Intel-based
models can run Snow Leopard (retail DVD) 10.6, & get a combo update
download 10.6.8. A RAM upgrade may be necessary for best performance.
So you could reply to this if so, then more info could follow.
Good luck & happy computing!