Q:

What prompts Aranea to join my party?

I am at the end of chapter 8, doing a good amount of side content before finishing the story. I am running towards a quest location and some daemons want to pick a fight with something that isn't their own size. 
After a few seconds into the battle, Aranea shows up out of the blue, joins my party, and helps me fight the daemons. After the fight, Noctis thanks her and she leaves.
What prompts Aranea to randomly join my party for the fight?

A:

Aranea Highwind's appearance in battles after Chapter 7 isn't triggered by any specific conditions.
Fandom states:

After the party's first encounter with her in Chapter 7, there is a
  rare chance Aranea will temporarily join them during battles on the
  open world. Her drop ship will appear near the battle and she'll jump
  into the fray, serving as a guest party member for the duration of
  combat. Once the battle is over, she says some parting words and jumps
  into the air, vanishing from the scene.

There is a glitch that remains in v.1.03 where you can force her into your party, at the cost of an ability to fish.

