Katniss' pregnancy with her son was easier than her daughter's. He has Peeta's blonde hair and Katniss' grey eyes of the Seam. He is a toddler during the epilogue, and seems to be at least few years younger than his sister. The epilogue also states that he has chubby legs. Katniss probably had him when she was around 32, as eighteen years ago, she was 17, and she had her first child 15 years after that. He does not know of his parents' roles in the Games, as Katniss does not yet want him to know. However, she does recognize that he will be taught as he matures. He is mentioned in the epilogue of Mockingjay chasing after his sister in the meadow on "chubby toddler legs", so he is about three years old.
As Peeta is a baker, it seems likely that Rye is named after the type of bread. Another contributing factor to this choice of name could be that Rye is only one letter off Rue, the ally of Katniss who died in the 74th Hunger Games.