What qualities do you look for when hiring for an executive operations position?

Having been on both sides of the coin, I would say trust is the first factor. Mentor the one with less experience and as he develops, your time will thereafter be freed up to work on long term strategic goals & objectives.

I would try to hire into your weakness and might take an inventory of exactly what you expect the candidate to aggressively deal with that will help your company scale. It seems probable that you are hiring your ops employee before an HR employee, and finance employee so if that is the case you will want to make sure the Ops candidate you select has the ability not only to just handle those roles but to also transition those responsibilities as the company grows. When I think of ops I think of day to day management but also strategy.

Promoting someone from within might be an easier and more comfortable route but an external candidate will most likely challenge you in the right sort of ways and compliment you in others, this uncomfortable chafing will lead you to be a better leader and strategist within your business. Good luck!

I'm working directly with the SVP of Operations of my company & based on that I would HIGHLY value a few "off the radar" behaviors; the ability to quickly adapt to change, continuous learning, strong implementation skills, and the ability to identify/develop talent.
You're welcome to contact me for more detail - not knowing the experience of your internal candidate, hard to answer that part of the question.