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.