Scroll down to the picture of the lute bowl. You should be able to see the end pin (strap button) at the very bottom of the lute bowl. It lies 1 inch or so below the level of the soundboard.
There are two types of strap buttons. The type with a metal screw are normally used for electric and acoustic steel string guitars.
The other type is entirely made of wood and has a tapered shaft. This is the type used for nylon classical guitars and the type for lutes is the same but smaller. Ideally you need a reamer to fit this type of strap button, which is hardly a tool that many people have lying around.
It is possible to improvise if you have a bit of experience with this sort of thing.
As for drilling the hole. Be very careful, you need a very sharp, very good quality lip and spur drill bit. I use an eggbeater drill (the type you turn by hand). That type of drill allows a lot of control, so you don't burst through the inside of the lute. You have to go very slowly and apply very little pressure. The idea of using a Dremel (that someone posted earlier) might not be the worst idea. It's almost certainly a better idea than using a poor quality drill bit that has been put into a 18V cordless drill.
Edit. The pictures posted by Cadiz show the position much clearer.
I recognise those lutes. The Early Music shop are now offering a different series of lutes, what they term the 'heritage'. I've yet to see one in person so I can't comment on how good, bad or indifferent they are. They seem to have a laser cut rosette which is hardly surprising on a cheap lute. I doubt such a rosette will affect the sound much. I'll probably get to see one within the next few weeks. They also do a kit version for those who feel adventurous and who like gluing things together. In which case mounting a strap button might be the least of your problems. You'll save around £300, although I suppose it could cost you £900 worth of grief.