If you haven’t had the conversation with your relative about getting those “unofficial” tenants out, I believe you should do that soon and present your plan how it will be beneficial for both of you and you can begin the process of getting new tenants in. 

But if you already own the home, if you have a skill set that compliments theirs or the downpayment money, their experience may be leveraged to finance the deal.And there is always the "friends and family" approach of trying to get one of your relatives to put up the money, either as a loan or equity, to get started.  

Unless you can convince the lender to do a seriously low short sale, I'd take a hard pass on your relative's problem.  

If you buy your relative’s house for $130k-$140k and put in $10k of updates, make sure to get a Broker Price Opinion (BPO) and/or a formal appraisal to show its true value.

If you can increase the rent enough to get a good return, then perhaps it is worth keep but I doubt that will go over well with your relative

Use your relative’s money for the down payment. 

I agree with the others that you can create a written lease that states she is the sole occupant, then kick your relatives out if they try to occupy.

So here is what you can expect with this deal:1) You won't want to let any lenders know that this is your relative. 

You have (presumably) paid the taxes and allowed your relatives as your guests to be on the property for purposes you agreed are voluntary and can be revoked at any time.

Your relatives probably don't mind because they don't pay state income tax.