Here's why Han Solo and Princess Leia naming their son Ben doesn't make sense. Introduced in the Star Wars sequel trilogy, the villain Kylo Ren is eventually revealed to be Ben Solo, offspring of Han and Leia. Born shortly after the Galactic Civil War, Ben trained in the ways of the Force at Luke Skywalker's Jedi Temple before he turned to the dark side and became Supreme Leader Snoke's student. As he grew older, Ben resented his family's perceived weakness and instead tried to embrace a destiny as Darth Vader's heir apparent.
Audiences first heard Kylo Ren's birth name towards the end of The Force Awakens, right before Kylo killed Han Solo. The name Ben had been used in the Star Wars franchise earlier, as it was Obi-Wan Kenobi's chosen alias during his exile on Tatooine. While this could have been a coincidence, Star Wars comics have revealed it isn't. Ben Solo was named after Obi-Wan Kenobi, a fact he hated since he felt like he was supposed to live up to the promise of an ancient Jedi he never knew. And it is a curious on the part of Han and Leia.
While Leia does exclaim "Ben Kenobi, where is he?!" after Luke rescues her in A New Hope, she primarily knew the Jedi as Obi-Wan or General Kenobi, presumably from stories Bail Organa told her. Han, on the other hand, barely knew Obi-Wan at all and really had no personal connection with him in the short time they spent together. A case can be made fatherhood made Han more sentimental, but Ben is still an odd choice for his son's name.
Naming Han and Leia's kid Ben seems like Lucasfilm's attempt to pay homage to the disbanded Expanded Universe, where Luke had a son called Ben. However, there it made more sense given the dynamic between Luke and Obi-Wan. They were clearly very close, as Obi-Wan was watching over a young Luke well before he started training Luke in the ways of the Force. Additionally, Luke knew Kenobi as "old Ben"; there's only one instance in the original trilogy where he refers to his teacher as Obi-Wan. It was fitting when he decided to honor his mentor in the EU since there was real meaning behind the name selection. Besides Uncle Owen, Obi-Wan was the closest thing Luke had to a father figure, a claim neither Han nor Leia could make.
An argument can be made that the name Ben Solo is a touching gesture, since Obi-Wan helped bring Han and Leia together in the first place. If he hadn't hired Han to fly him and Luke to Alderaan, Han and Leia's romance probably never would have happened. But given their personal histories with Obi-Wan (especially Han), even that feels like a stretch to use him as their child's namesake. And it ultimately didn't do Ben any favors. He was under enough pressure as the descendant of famous galactic heroes who helped defeat the Empire, naming him after a Clone Wars general made that burden even worse and indirectly led to the events of the Star Wars sequels.