Any poorly bred dog can be prone to health problems, whether it is purebred or mixed. If the breeder knows nothing about the genetic health and temperament of the dog's ancestors (including parents, grandparents on both sides, for several generations), neither the breeder nor you have any way of predicting how the puppy you buy will turn out.
This is the case with backyard bred ("cheap") purebreds, as well as the mutt that your friend has.
If you decide to go for a good old mutt, I'd recommend getting one from a shelter or rescue rather than encouraging your friend to breed their mutt. That way you will be saving a life, rather than encouraging bringing more questionably bred mutts into world for no reason.
If you decide to go with a purebred, I'd encourage breed-specific rescue, or saving up for the cost of a well-bred puppy from a responsible breeder that knows the pedigree of the dog, and does the proper genetic testing.
ADD: Understand that most of these people are simply wrong, and there is no such thing as "hybrid vigor" in the mixed breed dog. The fact is, if you breed two poorly bred dogs carrying genetic health problems, those problems will show up in the puppy *regardless* of whether that puppy is purebred or mixed. Having worked in the veterinary field for many years, I have just as much anecdotal evidence of mutts with genetic health problems (hip dysplasia, skin allergies, and thyroid problems being chief among them) as healthy ones.
If you don't know the genetics behind the dog, you cannot predict the health, temperament, or mental stability. It's really that simple.
Oh...and ignore the fool who posted the link to the BBC video - AR propaganda at its best.