El Duque

You will need to explain how Ms. Miller is attempting to thwart the democratic will of the voters. Her case doesn't suggest the referendum vote was illegal or even trying to stop Britain from leaving the EU.
What her case does is ensure a proper process for Britain to leave by ensuring Parliament which is the legislative branch makes the decision rather than the executive branch through the Prime Minister. 

Given that so many Leave supporters were complaining about having no say about decisions made in Brussels that impacted them, they would be quite supportive of Ms. Miller's approach which ensures the legislative branch which is representative of the people is making the decision rather than the executive branch which could be considered part of the "government elite" .  

The referendum was about Britain taking back control of policies to give its Parliament more decision making. Ms. Miller's case is very consistent with that approach.