You did not post enough facts to provide a complete answer. Texas law has small estate and summary administration proceedings for estates of less than $10,000 and $50,000, respectively. What process you use depends on the value of the estate.
You should contact a probate attorney who can help put your father's mind at ease. The attorney fees for representation in such cases are generally very nominal.
Some banks will allow the closest relative to access a safe deposit box even if the relative is not a signer on the box if the relative tells the bank that he or she is looking for will(s). If the estate does not qualify for the small estate procedure, and when an attorney has filed a petition with the court and gotten an order appointing your dad as representative of your mom's estate, those papers should be taken by your dad to the bank to gain access to the box.