In Interstellar Tom, in his record, says that he wanted to name his first kid (Jesse) 'Coop' but his wife said "next time." But Cooper is a last name in the movie, not the first name of the character. Why didn't Tom name his kid 'Joe' (Cooper's first name) if he wanted to name him after his father? Also, wouldn't that make Tom's kid's full name 'Coop Cooper'?
Evidently his wife agrees with you that it's a pretty dumb name so she (quite rightly) put her foot down, although in the end he did get his own way since that's the name of their second child
“I wanted to name him Coop, but Lois said maybe the next one. Grandpa said he already had the ‘great’ part,” Tom went on, “so we just leave it at that…”
As to why he wants to call the boy 'Coop' (and not Joe or Joseph), it's down to the fact that he wants to honour his father by naming the child after him and that's the name everyone, including his son, seems to know him as.
Out of universe, the explanation is rather simpler. His name hasn't actually been revealed by this point in the film, and since the station isn't named after him in the revised version of the script, there's no need to 'call back' to his name later in the film. As far as we're concerned he's just plain old "Coop" to his friends and "Mr. Cooper" to those who don't know him well.