How to decide on the spelling of Antoinette's name

Antonette T. GEBAUER Wallace, a great grandmother on my mothers side, she was my grandpa Wallace's mother. I have found her name spelled; Antoine on a ships passenger list with her parents, Antony 1880 census, Antonella 1900 census, Antoinet 1910 census. But on a photocopy of my grandpa's social security application he wrote his mother's name as Antoinette T. So for now I will spell my great grandmothers name Antoinette.