This is actually a really tricky situation especially given that you've paid off the mortgage.
Absolutely do not charge him rent without getting legal advice first. Remember you'll have to pay tax on any rental income over a certain period and more importantly, it would make your relationship into a landlord and tenant, or a lodger, relationship and this could cause issues if the relationship suddenly ended and you wanted him to leave. If you didn't register officially as a landlord but he still paid you money that could be seen as rent, you could be in a lot of trouble if it was suddenly discovered that you had undeclared income. I don't think he would be entitled to a stake in the property though as there's no way he could claim he had contributed to mortgage repayments etc.
Honestly, If I were in his case, I wouldn't be willing to pay rent to a partner if they had already paid off their mortgage and were essentially making a profit from being in a relationship with me. I'd feel I was being taken advantage of, especially if the area/property was a nicer and more expensive area than what I would have chosen for myself.
If I were going to pay rent, I'd want to ensure I had rights equal to what I would have if I were renting an actual property (e.g. a lease, notice period, right to complain and have repairs done etc.).
But then again, I know I would also feel resentful if I were letting someone live rent free in a property that I had paid for.
Might it be better for him to move out into his own place? Or he tries and buy a place which he then rents out and you then share the rental income?