She knows Obi-Wan's real name as well as his alias of “Ben.” When Luke comes barging into her cell and says, “I'm here with Ben Kenobi,” Leia jumps off the bunk in an instant. “Ben Kenobi, where is he?” she says, raring to go. That name was all she needed to hear. Not Obi-Wan.
Prior to this series, it never made a lot of sense why Leia would've named her son after Ben Kenobi, the mentor of her brother whom she never met.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
Throughout her encounters with Kenobi in the series, he refers to himself as Ben. The most straightforward explanation would be that Leia knows the person who rescued her is Ben, and never connected the dots that Ben is actually Obi-Wan.
In A New Hope, Leia's utter faith in Obi-Wan Kenobi comes circuitously from her father, who, in that film, is never glimpsed. However, when Luke breaks Leia out of her cell and says, “I'm here with Ben Kenobi,” and Leia doesn't miss a beat. She knows that's his other name.
Leia did remember him, that's why she knew where he was (Tatooine) and the general area, and knew his real name, not the 'Ben' one. This message was of utmost importance so Leia used her senatorial voice to convey the need for urgency.
“So Leia Force projects herself as Ben and uses her life force to save Rey. Even going as far as to kiss Rey to convince her it's Ben. It's why when Ben disappeared, so did Leia.”
The crux of the series is that Obi-Wan must rescue Leia from the clutches of the Empire. During this series, her bond with Obi-Wan, or “Ben,” results in something surprising: It legitimizes the naming of Leia's future son, Ben “Kylo Ren” Solo.
In Death Star, it seems the Sith Lord could have discovered Leia's true identity if he'd pursued his hunch. However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover she was his daughter.
Darth Vader only learned he had a daughter in Return of the Jedi - but there's no evidence he realized she was Leia before his death and redemption. Darth Vader may have never known Princess Leia was his daughter in Star Wars and never cultivated a relationship with her, even when he was a Force ghost.
The reason Darth Vader could sense Obi-Wan and Luke was because of how they actively used the Force several times throughout the Original Trilogy. Vader never sensed Leia because she didn't the Force; Leia did not even know she had Force powers to begin with.
On Tatooine, the droids are captured and sold to Luke Skywalker's uncle. As Luke is cleaning R2-D2, he accidentally triggers part of the message from Princess Leia, which plays on repeat. "Help me, Obi-Wan Kenobi," she says, "you're my only hope."
We know from Leia's parents and from moments in the Star Wars sequels that she was, in fact, Force-sensitive, even though she didn't get a chance to display those powers in the original trilogy.
She's important in the plot sense because Reva, who is searching for Obi-Wan for Anakin, uses Leia as bait but she's also important because she reminds Ben what he lost in Padmé and Anakin but what he gained in the twins. Leia is just as important as Luke.
With Leia, Luke, and Darth Vader all being major characters in "Obi-Wan Kenboi," the show is having to tread lightly as the "Star Wars" canon is pretty explicit. We know exactly when Vader discovered that Luke was his son, and we know when he figured out that Leia was connected to him as well.
She doesn't bat an eye, shows absolutely no emotion. Luke is devestated but she is cold, like she never knew Obi-Wan.
Obi-Wan Only Says He Never Owned R2
Note that he says nothing about not recognizing R2; his clever dodge gives the impression that they're strangers but doesn't actually deny knowing the little astromech. Thus, there really isn't a "contradiction" in the first place—Obi-Wan was just being his cunning self.
This is not because the Third Sister knows that Leia is Anakin Skywalker's child; it is because the Third Sister found a link between Obi-Wan and Bail Organa and she believed Obi-Wan would do anything to save Bail's daughter.
Has Leia 'forgiven' Anakin Skywalker for his crimes as Vader? Yes, in the “Legends” part, she even named her third child for him, as a symbol of his heritage and redemption. However, it's more than acceptance that Anakin is her BIOLOGICAL father and she got her personality and Force sensitivity from him.
While other factors (like her love for Han) kept Leia from becoming a Jedi, Darth Vader's evil existence was definitely a big part of it. Leia wanted to lead and reshape the galaxy, but wielding the same power that had corrupted Vader would be too big of a temptation because she would have become a powerful Jedi.
In the post-2014 canon universe, the story that reveals Darth Vader discovering Luke Skywalker's identity (and thus the fact that the Death Star's destroyer is his son) is issue 6 of 2015's Star Wars comics by Marvel, which takes place sometime shortly after A New Hope.
The simplest explanation is that Obi-Wan is using Ben as an alias, so that he can hide his identity from the Inquisitors and the rest of the Empire. Even if there are deeper reasons for the name, which we'll get into in a minute, this is most likely a large part of the reason.
In the original Star Wars: Episode VI – Return of the Jedi screenplay, Obi-Wan did have a brother, and it was none other than Owen Lars (Joel Edgerton)—Luke Skywalker's step-uncle and Anakin's step-brother. This detail was cut and did not make it into the film.