HOW CAN I FIND OUT WHY SOMEONE ..WHOM IS A NON FAMILY MEMBER IS USING MY LAST NAME AND HAS MY FATHER LISTED AS NEXT OF KIN...on public records

It isn't illegal per se for your father's ex girlfriend to use your last name, depending on what she's using it for. What kind of public records? When you received a call about refinancing your father's property, was her name mentioned, or was it a random telemarketing call? Regardless, I would be concerned as well. First, ask your father whether he made any transactions with this ex, such as putting her name on the deed to his house, lending her money, making her power of attorney, or making a will and putting her in it.  Then, whatever his answers, you can't be too safe. Have your father make a new will, and a power of attorney, and do a deed and lien search on his house to make sure this unscrupulous-sounding ex has not falsified documents to enrich herself.  Finally, depending what you find (or to handle all of this), you'll want to hire an attorney to contact this ex, minimally to advise her to stop using your name to falsely claim to be next of kin to your father, and if necessary, engage in other litigation.