When the State Military invaded Rosé Thomas' hometown of Reole, she was supposedly raped and became pregnant, though this is strictly only in the 2003 anime canon. It is unknown who the baby's father is, what its name is, or even if it is a boy or a girl, though it is presumed to be a boy.
He first appears near the end of the series as a newborn wrapped in a blue blanket that is always carried in Rosé or Lyra's arms, usually seen sleeping or crying. When it is revealed that Lyra is really Dante, Lyra takes the baby and threatens to feed him to a monster. However, he is saved by Edward Elric and returned to Rosé. His crying notably causes the Homonculi Wrath to become distressed.
The last episode shows Rosé's baby a few months older in the Elrics' hometown of Resembool (possibly just coming for a visit). He is shown to be able to laugh and hold a bottle by himself.
Rosé's son is shown again in Fullmetal Alchemist the Movie: Conqueror of Shamballa. He is about two or three years old and back in Reole. When the Thule Society's suits of armor invade, Rosé rushes to his rescue and holds him close. It is shown that she has two other children now: a little girl and another little boy, both about the same age as her son. It is unknown where they came from, but that they seem to live with Rosé because the credits shows Rosé at her house swinging the little girl around while the two boys play together. It is presumed that they were orphans that she adopted since they immediately ran to her when they were frightened.