Ben Skywalker is a human male, the son of Luke Skywalker and Mara Jade, born in Greg Keyes' The New Jedi Order: Edge of Victory II: Rebirth (Del Rey, 2001) during the Yuuzhan Vong War. He is named for Obi-Wan Kenobi.
Born into a war
His arrival is written as a deliberate note of hope in the bleakest stretch of the Legends timeline — a war that had already killed a Solo child and depopulated whole worlds. As a small boy he shuts himself off from the Force entirely, a reaction the books treat as trauma rather than mystery, and coming back from that is his first real arc.
The apprentice
In Legacy of the Force he is taken as apprentice by his cousin Jacen — and the series uses him as the reader's instrument for watching Jacen fall, because Ben is close enough to see it and young enough to be talked around. Extracting himself is the hardest thing he does. The Fate of the Jedi series then pairs him with his father on a long investigation, and the two-hander is some of the warmest writing Luke ever gets.
Legends only
Ben Skywalker exists only in Star Wars Legends. Current canon has no child of Luke Skywalker; the name survives, transferred, as Ben Solo. Readers should not treat one as a version of the other.
Reading note
He is the strongest evidence that the Legends shelf was building a genuine multi-generational family saga rather than a series of adventures. Whether it stuck the landing is a separate argument.