My parents received money from my grandmother to put a down payment for a house. After my parents purchased the home and put the house under their names, my grandmother moved in and eventually after 3 years she moved out. Now she wants her money back which my parents can't pay for. She suggested refinancing the house, but my dad doesn't want to do that. He also fears that she might try and sue for ownership of the house. If she tried to sue, can she win the house even if we have proof that she authorized the fund transfer and that the title is under my parents name and not hers?
In the US (and I'd imagine elsewhere), before the bank gave your parents a loan to buy the house, they need to underwrite that loan. Underwriters are (generally) required to document the source of down payment funds. If there was a large transfer from your grandmother to your parents, that would have shown up on the bank statements your parents submitted as part of the loan application. The underwriter would then have required your grandmother to provide a letter indicating that the funds was a gift or to provide some documentation for the loan so that the additional debt could be counted properly when qualifying your parents for the loan (if it was a loan, the repayment of that loan would be part of the total debt load the bank had to consider to ensure the mortgage could be repaid). Assuming this happened, there should be something in the paperwork from your parents' loan application that would specify whether this was a loan or a gift.
If your grandmother signed the letter indicating that this was a gift (which is normally how this sort of thing happens), she can't legally sue now to recover the gift (well... she can sue, she just won't win).
Practically, if this was really a gift with the understanding that grandma would live in the house and that is no longer a reasonable option, your parents may want to pay back some or all of the money in order to maintain family peace and relationships. But legally, a gift comes without strings.
Note: If your parents got their mortgage back in the 2006-2007 time frame, it is possible that they got one of the NINJA (no income, no job, no assets) that were frighteningly common so no underwriter bothered to ask. But if they took out a loan in the US after 2008, they almost certainly went through the underwriting process and the underwriter almost certainly documented the source of funds.
My parents received money from my grandmother to put a down payment for a house.
This sounds as if it was supposed to be a gift. Or was it only loaned?
Now [grandmother] wants her money back which my parents can't pay for.
On what grounds does she want the money back?
She suggested refinancing the house, but my dad doesn't want to do that.
If it was a loan, she can request it back. Then your dad might have to get a loan or to sell in order to pay her out.
He also fears that she might try and sue for ownership of the house.
She can try to get a lien on it and then eventually seize it, but that would be a dick move.
If she tried to sue, can she win the house even if we have proof that she authorized the fund transfer and that the title is under my parents name and not hers?
That completely depends - in the first step, a court might decide whether the money was a gift or a loan.
If it was a gift, you win.
If it was a loan, your dad has to pay it back under the defined conditions. Or, if no conditions were defined, under conditions which might be set by the court.
If he has to get a loan from the bank, he should do so. If he can't, or doesn't, grandma might want to have the house seized, but that might be hard or even impossible.
Disclaimer: IANAL and laws might be different.
He also fears that she might try and sue for ownership of the house. If she tried to sue, can she win the house even if we have proof that she authorized the fund transfer and that the title is under my parents name and not hers?
This skews more towards the legal side, but generally she's not going to be suing for ownership of the house. The legal term here is lien. If you buy a car and finance it, both you and the lender will get a car title so they can repossess the car legally. Most states even have slots for multiple lien holders on them by default. Houses are quite a bit more complicated than that, but they generally work the same way. If someone has a vested interest in the property itself, they'll have some sort of legal document showing their lien and will file court briefings indicating why they need repayment or ownership (i.e. the borrower defaulted).
If your grandmother wanted to pursue legal action, she will almost certainly sue your parents directly for the money she gave them (which is something she, in theory, has legal standing to do). If your parents lost that that lawsuit they could sell the house to pay any legal settlements made against them. In other words, nobody gets the house, only money.
There is not much to add to Justin Cave's answer but I did have an additional question. Did grandma pay rent during the three years she lived with them? If not, or it was below market your parents could make the argument that the rent counted against that loan.
So really this is a long uphill battle for Grandma. She has no documentation for a loan and even if there was documentation, she potentially received room and board in lieu of payments to that loan.
Most lawyers would consider this a waste of time, so perhaps the only recourse is small claims.