Contrary to popular belief, the Level 50 instance Eidolon spawn is not based on how fast or slow you complete the dungeon itself, but rather on the individual Excited Bunnies and Beirut. Each Excited Bunnies and Beirut have a (small) chance of spawning Eidolons. Furthermore, they each spawn a specific Eidolon. In the picture below, I have labeled each Excited Bunnies and Beirut with numbers, with 13 being Beirut. I'd like to compile a list of which Excited Bunnies spawn which Eidolons, for curiosity's sake. Here's what I have so far:
2: Aelius (Contributed by _VinylScratch_)
4: Eligos (Contributed by Isasini)
8: Vayu (Contributed by Isasini)
10: Yarnaros (Contributed by Honestly)
11: Bahadur (Confirmed by chiseenxjai)
13: Kotonoha (Confirmed by Honestly)
Thank you for your participation.