Tragedy literally means
a goat's song or ode.
Is there any indication in mythology of why a goat song would be sad or dramatic?
As far as I know, Pan, Faunus, Silvanus... all part goat deities are more in tune to celebrating life to the hilt...
The term was explained by ancient writers as "song for [the prize of] a goat" or "song for [the sacrifice of] a goat".
Walter Burkert observed in Savage Energies: Lessons of Myth and Ritual in Ancient Greece that these were hard to disentangle, because the song competition would mean competing for the honor of being given the goat that would be offered as the sacrifice to Dionysus, and thus for the honor of actually giving the sacrifice. He also pointed out that the "goat skin" interpretation does not work grammatically, the terms are inflected incorrectly.
Originally the song would be a choral leader and the chorus. The additional of a second leader and then a third allowed true drama. And it was drama rather than tragedy in the modern sense. Aristotle in Poetics described the best plot for a tragedy as one in which the main character was about to do something that, unbeknownst to him, was particularly dreadful, to learn in time that was particularly dreadful, and so to not do it -- that is, to have a happy ending.
The German Wikipedia article on tragedy explains this well:
The word "tragedy" stems from ancienc Greek theater and denotes the "he-goat's song" or "song for the he-goat's price" (gr. τραγωδία, tragodía). In the Dinoysos cult, a "Kosmos" (gr. κῶμος kōmos) was staged, a procession with song, the participants masked and wearing the skin of a he-goat (gr. τράγος tragos), to impersonate the god himself or his accompanying satyrs. The theatrical form of tragedy developed from a myth sung in chorus, ... The choral parts of the theatrical dramas are a rudiment of this ancient form, the dialog parts a later, secondary addition. (my translation)
No source is given on Wikipedia, but all this is in tune with what I remember from studying literature and learning of the origin of ancient Greek theatre.
A tragedy, in ancient Greek theatre, is not "sad" in the contemporary non-theatre-related usage of the word (e.g. "What a tragedy!"). The Greek tragedy is defined by a fated conflict of the protagonist: the failure of the protagonist is unavoidable, because of the constellation he or she are placed in. Think of Oedipus: He did nothing wrong, it was his fate to kill his father and marry his mother. In contrast to a modern hero, there was nothing that he could to to avoid that fate. Modern heros on the other hand either succeed or fail because of their own deeds.
The ancient Greek comedy also derived from the Dionysian procession (see "komos" in my translation above). But while the tragedy took the mythic part of the procession and told stories of gods and men, the comedy took the festive part and became a satire and critique of (then) current politics and society. The ancient Greek comedy was more like today's political satire, than today's comedies.