How do I "finish" basement with old stone walls?

We are renting my grandmother's home who is under hospice and though it is spacious, the house has space in all the wrong places. For example an enormous living room, but very very very small bedrooms (the master will barely fit a queen with 2 night stands, and MAYBE 1 dresser!!) ANYHOW because of this we need an extra bedroom for our oldest son and a game/workout room. There are 2 areas to the basement, the right side is cinderblock and that would be the game/family area which we could paint the cinderblock if we wanted to. BUT the area on the left is the 'original' basement which the walls are constructed of large stones, some of them stick out, they are uneven and a bit crumbly. How can we "finish" this part of the basement into our son's room? My husband can drywall and I have never purchased it before but it seems cheap enough, however can drywall go right over that or would we have to totally hire someone (remember we are renting!) Thanks!