The ambiguity of the Queen's Powers could be used to her advantage in the old tradition of the Crown siding with the populace, even though that populace be divided. I suspect that, were the Queen to take the initiative, on the advise of selected council, (her choosing between the various councilors on offer), and resolving the crisis on the basis of enabling parliamentary action, that such action would redound to her credit as perhaps the crowning achievement of her long reign.

The Queen would NOT be deciding the Brexit issue. She would, instead, be providing a way for the public and Parliament to chose. Likewise, a motion by Parliament to have the Queen send a letter on behalf of Parliament requesting an extension of the Article 50 deadline would probably be accepted by the EU. The very fact of such a letter would be irrefutable proof of profound change in the UK government from its current state and a way out of a stalemate for both the UK and the EU.
 

"It is not necessary to have hope in order to persevere."