The general rule is that boundary line fences are usually the joint responsibility of both neighbors. The fact that the fence is about 4" off your mom's property line may be enough for her to deny paying for part of the repair. But, it might not be enough to stop the neighbor's from fixing it and then suing her for 1/2 the cost.
It would be important to know why the fence was put where it was. Was that intentional, or was someone mistaken about where the property line was? Did your mother treat the fence as if it was hers? Did she have another fence on her side of the property line? Answers to those questions might give more insight on your mother's risks and obligations.