How can I change a last name?

I want to know the way I can change a last name. The name change would be made for my grandfather, who was recorded solely by his mother and not by his father. He never received the appropriate name of his father, which does not carry it but that is rightfully his. Both parents have died, but has several half-brothers that do carry the name that corresponds to my grandfather and are willing to help him to make the change, even if it requires a DNA test to check the relationship between them. This is why it would be useful if someone could provide me the procedure to follow to carry out the change. Thank you.

I read that can be performed:

CORRECTION OF MINUTES WHEN IT WAS NOT SEATED A FACT THAT SHOULD HAVE SETTLED (missing a name, a surname, etc.)

P.S: I HAVE LETTERS FROM MY GREAT-GRANDFATHER AND FROM MY GRANFATHER'S SIBLINGS TO MY GRANDFATHER.