Does my mother have the rights to her ex husbands home?
I would question if the second marriage is valid if he wasn't legally divorced prior to marrying for the second time.
I suggest speaking to an attorney in their state who is well versed on this topic.
Each State has it's own rules associated with death. Did your dad have a Will or do the interstate rules for you State need to be followed? Was her name jointly on the house "with rights of survivorship?" In that case, it doesn't need to be probated, it was her's by virtue of survivorship.
Was the marriage ever made legal after the divorce?
You may want to reach out to the divorce attorney that handled your mom's case.
I hope that you have been in touch with a lawyer because from your posting, it seems that your dad was still married to your mom at his passing. The best advice is to consult with a local lawyer. Probably not what you want to hear, but that is the soundest advice!
You may want to refer to the the deceased as your mom's late husband.