Surname problem of my 59 year old mother

Hi Po. I would like to seek for your advise regarding my mothers surename. Balak ko po sanang kumuha ng passport nya para man lng maexperience nya na magtravel abroad. Ito po concern ko sa name nya:

• Ang mother ko pa anak sa pagkadalaga ng kanyang mother.
  

• -All her documents and IDs (marriage certificate, SSS and school records)  my mother is using the surname of her step father.
  

• -But her NSO birth certificate record, her surname is the maiden name of her mother.
  

• -Both her mother and step father are deceased already.
  

• -My mother looked for SSS record of her step father but she was not included as beneficiary.
  

Question po:
1) Any other documents na pwede gamitin to justify that my mother can use the surname of her step father?Pwede po ba pictures or testimony from siblings who are real children of the step father?
2)Would it be better to use the surname indicated in the birth certificate and then just change the IDs and other documents (marriage cert)?
3)Ano po kaya pinakamadali na ways para makakuha sya ng passport.