Yeah it's absurd that she didn't change her name. Even if her parents (and their parents, and their parents) thought "well, we shouldn't have to change our names just because this guy in the past we never met was a bastard", La'an hates her name and got bullied for it, so the idea that she wouldn't just change it is comical. The only explanation is that she likes to play life on Hard difficulty and loves having to introduce herself as Hitler to everyone she meets.
To be honest I suspect the character was conceived with that name as part of Secret Hideout's hobby of linking absolutely everything back to an existing "iconic" Trek thing, hence why Burnham is Spock's sister and visits the Mirror Universe, and why Pike, Spock and Number One boarded the Discovery, and why Picard was whisked off to the past to meet Guinan with Q and Soong and Seven of Nine and the Borg Queen there, and why now in SNW we're fighting the Gorn and going to the Amok Time arena and hanging out with Christine Chapel. It wasn't enough to create a cool and tough new security chief, she had to be connected to Khan and be a previous captive of the Gorn. Her grandma probably got eaten by the Planet Killer while visiting Trelane on the Shore Leave planet. I'm just amazed they had the restraint to stop themselves making Ortegas, one of their few all-original* characters, the daughter of Richard Daystrom or something.
*I know, Jose Ortegas
The upshot is that SNW's rotation of weekly writers have so far shown talent, and so subsequent writers may well be able to create some fantastic stuff using La'an and her link to Khan. Plus, Chong has already emerged as one of the most impressive actors in the series, so she'll be able to sell just about any plot. But the basic premise of the character is pretty silly, yeah.