First, please note that I have not been able to find any canonical information on why Kylo Ren was given that birth name, so my answer is based upon reasoning and conclusions drawn from watching the movies (ep. 4-7).
Answer While it's true that Han doesn't seem to like Kenobi nor agree with him on anything, Kenobi did disengage the tractor beam and gave his life to ensure that Han, Luke and Leia could leave the Death Star alive and, relatively, unscathed.
If nothing else, Kenobi is the reason (though not intentional)
Han and Leia met (by employing Han to transport them).
Leia knew Kenobi to be a great hero of the Clone Wars, a wise general and a great jedi.
She likely held Kenobi in high reverence and would probably think that naming her son after him would be a) a great homage to the late Jedi and b) give her son a namesake to look up to.
scifi.stackexchange.com/users/33325/Gunnar Södergren
Although you may never hear this from Disney ever, it is very likely that Kylo Ren's real name is inspired by Legends' very own Ben Skywalker .
In Legends, Ben Skywalker is Luke's own son, and it was Luke who decided to name his son after the Uncle Ben who watched over him growing up.
Even though the Lars tried to keep Luke away from Obi-Wan because they did not want him to become a fallen Jedi like Anakin, Luke still managed to form a friendship with Obi-Wan when he could get away from the Lars' attention, developing an affection for the ageing man.
Later events in the Original Trilogy would only serve to deepen Luke's love and respect for Obi-Wan.
Kylo Ren was born 0-1 years after the Battle of Endor.
Leia had just recognised Luke as her brother and formally reunited with him.
Luke and Leia both greatly admired Obi-Wan (Uncle Ben to Luke and General Kenobi to Leia), so the two of them probably came to an agreement among themselves what to name Leia's son (I guess Luke's vote counted more than Han's), just like how it happened in Legends for Luke's own son.
But why Ben as opposed to Obi-Wan?
This quote from Legends' Luke may hint at an answer.
"I'm just afraid of placing an extra one on his shoulders.
Suppose I named him Obi-Wan, as a salute to my old master?
Would he think that means that I want him to grow up to be a Jedi?
Would he think he had to live up to Ben's reputation?
"
This line of thought aligns well with Disney-canon-Leia's mentality that she wouldn't want to over-burden her son.
Conclusion :
Drawing parallels from Legends, I believe Luke had a hand in deciding on Kylo's birth name, and the above quote explains why Ben instead of Obi-Wan.
Disclaimer :
My answer is based fully off of Legends material, as your question cannot be answered with Disney canon sources alone.
However, given that the name Starkiller is drawn from Legends, it is not the first time Disney drew inspiration from Legends without explicitly admitting to it even though it's blatantly obvious.
Given that this is also a case of a son of the Skywalker line being named after Obi-Wan Kenobi, the similarities are plenty enough to believe that strong parallels had been drawn by Disney from Legends for my suggested answer to be reasonable.
scifi.stackexchange.com/users/44906/thegreatjedi
She trusted him enough to deliver the plans in A New Hope.
Which he succeeded in doing by sacrificing himself which allowed Leia and Han to escape with their lives ...
I was named after a TV character from MASH...
scifi.stackexchange.com/users/59824/ShameWare
I'll posit another possible reason.
Leia makes clear in the movie that she was worried about Ben/Kylo going over to the Dark Side - perhaps from the start.
It was always a risk with Skywalkers, after all...
Naming him Ben may have been in the hope that it would help keep him on the Light Side.
People often name their children after famous people, in part in the hope that some of the virtues of the famous person carry on to their child - perhaps because the child looks up to their namesake, or perhaps for other reasons.
Naming him after a famous Jedi Knight who was always true to his word and was a great warrior for the Light Side may have been an attempt to guide their son to follow in his footsteps.
Of course, it didn't work out very well...
scifi.stackexchange.com/users/12858/Joe
As thegreatjedi notes, his answer is non-canon.
Gunnar's answer is missing some key points in my opinion.
Yes, it's true that Leia never met Kenobi and that Han met him for a very short time, and didn't really like him.
However: As, Gunnar pointed out, Leia knew him as great hero of the Clone Wars At the time Han knew him, he was still in smuggle mindset and not committed to the Rebels
But after the events of A New Hope , you have a span of years where Leia and Han are both fully committed to the Rebel Alliance, and are working closely with Luke.
And we know Luke holds Kenobi in extremely high regard and as a father figure.
Then, even later (after Return of the Jedi ) Luke founds the new Jedi Academy.
So basically the entire success of Rebel Alliance, the downfall of the Empire, and the reason there are Jedi in the galaxy at all can be attributed partly to Kenobi.
Also, why would one assume you have to know someone well to name your child after them.
People do it all the time.
You just have to revere them or their ideals.
So, in light of all this, there's nothing surprising about Han and Leia naming Ben Solo after Kenobi.
Perhaps the only question is why they didn't name him Yoda Solo.
scifi.stackexchange.com/users/3823/ThePopMachine
It's not uncommon for people to ask for opinions/suggestions from relatives when naming their children.
Ben was obviously a very important person, a mentor and a father figure, to Leia's twin brother Luke.
If Luke didn't have children of his own (we'll have to see about Rey), then it could be a way for Leia to honor her bond with her brother.
scifi.stackexchange.com/users/None/
Kylo Ren’s real name is    Ben, and was presumably given to him by his parents, Han and Leia, in reference to Obi-Wan “Ben” Kenobi.   But his mother never met the person he was named after, and his father knew the person just for a short time, and wasn't too fond of him then. Why would they name their son that?
Why would Kylo Ren be named after this person?
scifi.stackexchange.com
<star-wars><the-force-awakens><kylo-ren>
scifi.stackexchange.com/questions/115863
scifi.stackexchange.com/users/6943/Stephen Collings
It is not clear exactly why Han and Leia named their son Ben, but there are some plausible theories. Firstly, although their bond with Obi-Wan might not have been particularly close, he certainly had a reputation as a Jedi and gave his life for them to escape the Death Star. Furthermore, it's possible that they felt that by naming their son after a virtuous Jedi, they might hope to steer him on a positive past and avoid him falling into the Dark Side. Views vary, the main ones being that Ben was named out of respect for Kenobi - either directly or in recognition of Luke's respect for him - or out of concern over what may become of him in his future life.
Kylo Ren was named Ben as a way of honoring the sacrifice made by Kenobi in giving how own life. The name was inspired by Ben Skywalkerl Because she trusted him to deliver the plans in A New Hope. Perhaps because she was worried from the start about Ben going over to the Dark Side. They named him out of respect for Luke's opinion of Kenobi - and their respect for Luke.