How does someone change the name on a birth certificate

I'm 55 years old and trying to get a passport.

The family name on my New Jersey Birth Certificate is different from the family name I've gone by my entire life. Apparently my mother was between husbands when I was born 55 years ago.

My driver's license, social security number, etc. all have a different name from my birth certificate and I would like to change it on my birth certificate.
All parents are deceased - Is this your question? Add additional information