Laura Ingalls Wilder did have a son, who was born on July 11, 1889. He passed away only 27 days later on august 7, 1889.
It is unknown if the Wilders named their son. The grave marker in De Smet, SD simply calls the baby “Baby Son of A. J. Wilder.” And neither of the newspaper announcements of his birth or death included a name either.
You can find more about Baby Son Wilder in his biography here, including copies of the birth and death announcements.