First thing to try is to hold down the Option key at boot. After a couple of seconds you should see a mouse cursor appear on screen, and very shortly after that an icon of the HDD. If you wait a little longer the install disc should show up. When it does, simply double click on that and it will boot from that disc.
If instead you see a password prompt, then you're going to need to take the thing somewhere to get that password cleared. You could potentially do it yourself, but doing this on the older style Minis that were vaguely cube shaped, is a bit involved and requires some tools the average person probably wouldn't have. There's also a decent chance of damaging something if you're not careful about it. So probably best to take it somewhere if you run into this situation.
If you don't see a password prompt, but after a time (less than a minute) the icon for the install disc never shows up, then you are probably looking at a bad optical drive. It could also be that you have the wrong disc in the drive, so double check that it is the correct disc for that system. It probably came with an OS and Applications disc, and you may have accidentally put in the Applications disc. If it is the OS disc, then you probably need a new drive, and it honestly may not be worth the expense.
A few years ago I gave my wife a Mac Mini. Before doing so I did set it up with a name and password of her choice. To my certain knowledge she never made use of this computer. in fact it was never taken out of its box.