I've been through this....
It depends on the judge. He/she may ask you why you want your child to not have his name....why yours. In turn, the father will have to give his reason for the name change. How old is your child?
Say that your last name is Smith, dad's last name is Johnson, and your child's legal name right now is-- Tommy William Smith
The judge could always hyphenate the last name: Tommy William Smith-Johnson
Or, put dad's last name as the middle name: Tommy Johnson Smith
Or, this next one would really be in your favor if you *had* to put dad's name in there: Tommy William-Johnson Smith
That would be his legal name, BUT any documents that you have to fill out (ex: school records) that ask for First, Middle, Last would drop that dad's last name off the middle....and would be Tommy William Smith (as he was named originally).