It can be a bit confusing at the start of the film, but does Han Solo know Kylo Ren is his son?
Han Solo does know Kylo Ren is his son. He tells Leia that he saw “our son” after he witnesses Kylo Ren carrying Rey back to his ship.
Later in the film, Han refers to Kylo as “Ben,” being the boy’s real name.
Han sees Kylo Ren, knows it’s his son
In Star Wars: The Force Awakens, Han Solo brings Rey, Finn, and BB-8 to Maz Kanata. The group needs a “clean ship” before transporting BB-8 to the New Republic, and Kanata can help.
As the group enter’s Kanata’s business, a droid relays to the Resistance that their missing droid has arrived. Simultaneously, a First Order spy informs her group that BB-8 has entered the location.
At this time, Finn admits that he’s a runaway Stormtrooper. Not wanting to involve himself with a skirmish against the First Order, he strikes a deal with locals who can transport him to the Outer Rim. Rey, who feels drawn to the Kanata’s basement, makes her way there.
While in the basement, she makes physical contact with Luke Skywalker’s lightsaber hilt. The hilt triggers flashback memories of Rey being left behind by her family, revealing the reason as to why she wants to go back to Jakku so badly.
Denying her connection to the Force, Rey runs away into local woods. While there, the First Order reaches Maz Kanata’s secret location and begins dropping troops.
Kylo Ren lands upon the spot and hears from a stormtrooper that Rey and BB-8 escaped. He heads into the woods to find them as Han, Chewbacca, and Finn are arrested. The group is quickly freed amongst the confusion when Resistance ships touch down on the scene.
Ren uses the Force on Rey to extract a memory of the map leading to Skywalker from her mind. He then waves her unconscious and carries her to a First Order ship.
As Ren carries Rey to the ship, Han sees the two. Once the battlefield is clear, General Leia Organa exits and greets Han. Sadly, Han says he saw “our son,” referring to Kylo Ren. He states firmly that Kylo Ren was here.
There are many more moments throughout the film that Han confirms the identity of his son. Later in the film, Ren and Han have a heartfelt moment where he calls his son by his true name, “Ben.”
So that’s the answer. Han Solo does know Kylo Ren is his son. He tells Leia that he “saw our son,” after witnessing Kylo Ren carry away Rey to his ship.