Out of a/the hat
Definition 1 of 2
Pro Tip 1/3Random Decision
Used when a choice seems to have no specific system or reason. 
We couldn't decide on a restaurant, so we picked one out of a hat.
Pro Tip 2/3No Preferences
Suggests all options are equally likely or acceptable. 
Any of these plans could work, so let's draw one out of a hat.
Pro Tip 3/3Luck-Based Choice
Emphasizes the role of luck or chance in making a selection. 
Who will go first in the game? Let's choose out of a hat.