How cani change my baby's last name

My son is 11 months his dad didnt sign his birth certificatebut i still stupidly gave my son his last name. He hasnt been in the pictureand i want to change his lastname. I spoke to a lady from legal aid and she said i have to file a custody case first. And now im scared if i do he will try to get custodyand i dont want that. He use to threaten me while i was pregnantthat he would take my son from me. I do have a child supportcase i openedwhen my son was 1 month its been 10 months and they still haven't settledit. Is it possibleto change his last name without filing a custodycase ? I grew up having my dads last name and he left when i was 9 and i hated it i want my son to have mine please help.