I can't answer your question that is based on only three facts.

Fact 1: He is on the deed.

Fact 2: He refuses to relinquish.

Fact 3: His mother threatens to sue.

Three facts and a lot of questions that need answers to fill in the gaps.

Go back to Square 1: How did Dad get on his mother's deed in the first place? Was she forced or coerced? Or did she ask him to accept half interest and thereby be interested in protecting it as an inheritance to the family estate?

What happened to make G'mother so adamant that Dad relinquish his interest in the house? Has he remarried someone she doesn't like or trust? Has she begun to lose her mental acuity? Is he unreliable (e.g., drinks, gambles?).

Why would he give up his interest in the house? Is G'mother offering to buy out his share and he simply doesn't want to give it up? Is she afraid he'll put her into the street and take over the house? Is her judgment in decline?

I suggest you set up consults with several attorneys who are experienced in Elder Law. Find one who 'clicks' with Grandmother. Grandmother and that attorney can work toward protecting Grandmother's assets so she can relax about being old, maybe a bit unaware now, but safe from being taken advantage of. [I'm not saying it's possible, but if she worries, it doesn't matter whether it's true or a delusion.] Also consider her possible point of view -- she worked to make a comfortable life for her family, and now she needs to be assured that she won't lose it. She might realize, or might not have a clue, that she needs some help in managing her assets. If she trust the attorney, she will be able to work with the attorney to protect her home and other assets.