I think just pitching up and looking around for a bargain is a hit and potentially miss approach. In October, many hotels, even if they have rooms, will not sell if they are under contract to a tour operator; so he would need to find one that is not bound to a contract. Just before Xmas, there will be far fewer hotels/accommodation available, and pre-booking would be advisable. At that time, there are unlikely to be real bargains owing to the relatively low supply of accommodation.
One euro a night? Bellevue in Alcudia offered this as some sort of special early this season, but it is hardly common. I'm not sure I'd ever take the word of a bloke in a bar.
Cheap accommodation is often available in hostels. Tourist offices can often help here, assuming they are open, and by no means all of them are after the season ends.
Apartments - maybe, but this time of year, owners are looking for longish lets, rather than a few days.
To be honest I think your mate would be advised sorting it out before he comes. One of the online hotel brokers might be best to try.