Rent during refurb?

[John Teachout]

We've been renovating a house that has extensive termite damage (not recent) and I'm working under the house replacing sections of joists that have been eaten up. Got a phone call today from a woman who is looking for a place to live and is familiar with the house since her parents live next door. She acts like she could live in it in the un-refurbed condition. It would be her and her 15 year old daughter. She would probably stay there 3 years as her daughter is starting her sophomore year. There are almost no rentals available in this county as the schools are desirable. I'm going show the house Monday and if she is ok with it, we would love to have someone in it. She is currently living next door with her parents so her kid can start school Monday in the county. My wife and I have discussed it some and are trying to decide how to rent it. We certainly wouldn't expect to get "market rent" for it in its present condition but I don't know that I want to give her a low priced contract and then scare her when the price gets jacked as renovations are completed. My thinking is to come up with a number and then give her a "discount" until certain aspects are completed. How would ya'll approach this situation.
The house is "rough" but livable. (my daughter stayed there several weeks recently). It doesn't have central heat or air, just gas space heaters and window ac units. There is a nice electric flattop range and a standard fridge. It's a two bed 1.5 bath but the tub/shower is in the bath off the bedroom. The house was built in 1900.
Thanks.