this is all presuming there are only the 2 brothers that were both children of g-ma AND g-pa and there are no other children of either g-ma or g-pa. That g-ma and g-pa were married at the time of g-pa's death and g-ma did not remarry. There is no lien on the property that needs to be dealt with. G-ma had no will. the house was held separately by g-pa (g-ma having no titled interest)
Grandfather dies and leaves house to 2 sons. One of the sons legally (deed changed) gives his half of the house to his mother and the other son keeps his 1/2 interest in the house.
the first thing you have to find out is how g-ma and g-pa held title. Was it only in g-pa's name at the time of his death? If not, g-pa could not give 100% of the house to anybody because he didn't own 100% of the house.
Now, since he "gave" the property to his sons, that means there was a will. Even with a will, a widow has a right to an elective share of the decedents estate (1/3) so, one needs to know if g-ma expressly consented to allow g-pa to give the home, in its entirety, to the sons. If she did and presuming g-pa owned 100% of the home, that would mean each son now owns 50%.
now, son 1 gave g-ma back his share. son 2 and g-ma own equal shares. When g-ma dies, barring a will stating otherwise and presuming the property is not required to be dealt with in some other way, her share (1/2 of the whole) would be equally split between her sons (each receiving 1/4 of the whole). That means son 1 now owns 1/4 and son 2 owns 3/4 of the whole.
now, to go further, we need to know where in the administration of g-ma's estate this is. Regardless of that, son 2 does still own his prior 1/2 interest so whatever happens, unless he agrees to sell his 1/2, it isn't getting sold. As to the remaining 1/2, if probate is still open, it can be sold (if you can find a buyer for 1/2 of a house shared with a stranger) or simply transferred to the heirs. That results in the 1/4-3/4 interest division.
what they decide to do at that point is up to them. They would both have possessory rights and they would both be liable for their share (1/4 to one, 3/4 to the other) of taxes and maintenance.
if son1 wants to sell, he would have to sue to partition the property (which usually gets real ugly with the owners of the property getting a bad deal but at least there is closure)
there are myriad other possibilities if any of the basic facts vary.