(written from a Production point of view)
According to the script, Henry was owned by Ensign Janklow. Janklow was scheduled to go to the Gantol excavations for three weeks, and was looking for someone to take care of Henry. Counselor Deanna Troi approached Amanda Rogers, who quickly agreed. While no mention of Henry was made during the final version of the episode, he can be seen sleeping on a couch behind Q, shortly before he and Amanda begin to chase each other around the USS Enterprise-D.