FunRoms are mentioned several times in Star Control II. The first reference is made by Fwiffo, stating that the "SOS message in alien language" from the Moon Base was nothing more than a Melnorme FunRom called "Winky´s Happy Night".
Another reference is made by "normal" Spathi captains, and refers to a FunRom involving a grand adventure through space etc., in effect being a reference to Star Control II itself.
Also, Tanaka claimed that his life was over, because in addition to his race being dead and his world being an airless rock, he had seen all of his FunRoms over a hundred times. This mood changed only when the captain told him that he had "something better than FunRoms" on board after returning from Admiral ZEX.
Considering this information, one may assume that FunRoms are the DVDs of the 22nd century and used to store both interactive and non-interactive content.