My husband doesn't believe me when I tell him our son will have his last name. The reason is because…

My husband doesn't believe me when I tell him our son will have his last name. The reason is because I was pregnant when we got married and still on my mom's insurance but if I changed my last name to his with social security then my mom's insurance wouldn't cover me anymore and at the present time we can't afford health insurance for myself. I keep telling him, me keeping my maiden name for now will not matter because we can still put his last name down for our son. I am correct right?