You are absolutely NOT being unreasonable.
Not only are you under no obligation to help your family members out in this way, you are also not doing her any favours. I was bad with money when younger and I only learned how to handle it by understanding its value and realising that if I fell no-one was going to pick me up. I cannot emphasise enough how important this lesson was.
Also, I take it this girl isn't related to your DH? Why should he have to leave with your family members? As you know from prev experience, living with someone you are not related to, or not in love with, is really hard and a pita. And there are possible complicating scenarios, like what happens if she gets a bf who is threatening towards your DS, or starts bringing hard-partying friends back to your house?
Last but seriously not least, it's not good for your relationship with DH to have a third wheel (not counting kids). It discourages intimacy. What happens if you want to cosy up (or more) on the sofa? Have a row? Wander around in your undies? It's clear that you are both massively looking forward to being a cosy unit in your lovely new house and I cannot see why you should put this girl's comfort before yours and your DH's comfort.
I can come up with about a million more reasons why this is a bad idea!
If you can't face the fall-out if you flat-out refuse, tell your family that the spare room is being done up so that you can rent it out to a real lodger as you have a huuuuge mortgage, then quote a price for this room that 'bad with money' girl would not be able to afford. Then spend a really, reeeeeally long time doing it up!