If you had no dinosaur monsters in the graveyard at all, I assume you wouldn't be able to activate Fossil Excavation in the first place. You can't activate something that will initially resolve with no effect.
However, let's assume you did have another dinosaur in the graveyard upon activation. You pay the discard cost immediately when you activate Fossil Excavation. By the time you have to select the monster, the dinosaur you discarded would be a perfectly eligible target.