Of the main characters seen here, who is the only one to have never had both their birth parents mentioned or shown by name, on screen?
The correct answer is Worf, who had his birth father, Mogh mentioned many times, but whose birth mother was never mentioned nor shown by name
All TNG principle characters who've ever had parents mentioned by name
Will Riker - Mother Betty
Beverly Crusher - Father Paul Howard, Mother Isabel Howard
Ro Laren - Father Ro Gale, Mother Ro Talia
In which episodes of TNG were these established? I don't recall any of them being mentioned in the series or films, which is the body of knowledge this "game" is limited to -- and by your own wording of the question itself.
Worf's mother is the perfect answer - I can't believe nobody thought of that.
I still think Data / Dr. Soong shouldn't count, as Data was built and assembled, not born from Julianna's reproductive organs, but whatevs.