Advise for unmarried mother to change son's surname to be the same as the mother

My son is now going 10 years old and I wanted to change his surname to be the same as mine. Unfortunately, the birth cert was over and done with very quickly when I was just 19 years old. The father has no proper work and was not able to give any financial support since my son started to go to school. I made it all possible by myself with the assistance of my family. Right now, I am working abroad and my son is left at my family's home. I want to have the possibility of taking him wherever I am so that I could take care of him personally.  Could you advise of what I can do?  What is the procedure I need to follow? How long will it take and the cost involved as well?