If you look at our discussions of genre for the first two plays of the Oresteia trilogy (Agamemnon and Libation Bearers), you won't find the word "comedy" anywhere. As we've been saying throughout this module, however, The Eumenides is really the play that sets the rest of the trilogy on its head.
The Eumenides shows how revenge doesn't always have to lead on to revenge, but instead can be brought to an end through legal justice and proper forgiveness. This makes it possible to have a happy ending—one of the most basic elements of comedy. Because the play still deals with serious, weighty themes, it still merits that "tragi-" part of "tragicomedy."
Because much of these themes involve conflicts between family members, or even discussions of what counts as a family member, it's also a "Family Drama." Finally, last but not least, the play is a "Drama" because it is a play. (You didn't think we'd let that one past, did you?)