Now, this isn't going to be a "Wave my magic wand and cancel Brexit" post. Because (a) that's a bit dull as a counterfactual and (b) while I think that having the referendum was a really, staggeringly, stupid thing to have done I also think that asking the public their opinion and then saying "And now we're going to completely ignore you" is how you destroy whatever remaining faith that anyone has in the political system, and push people into violent response instead.
So I'd hold another referendum. But I wouldn't just re-run the previous one, because frankly I reckon that doing a stupid thing twice, because you think that the second time you _might_ win is just asking for trouble.
What I'd do, instead, is take a leaf out of the New Zealand Referendum Approach:
1) Ask the public if they want change.
2) Ask them what kind of change they'd like.
So, having ascertained that the public were unhappy with the current relationship with the EU, and having given them some time to examine the fallout, I'd set up a referendum with multiple options, using instant run-off voting to let people fully express what outcome they wanted.
The options would be something like these ones, and people would be asked to rank them in order of preference. The option getting the least votes would be discarded, and all of the people who voted for it would get their votes moved to their next preference. Repeat until someone gets at least 50% of the vote.
This would mean that you wouldn't have a "Leave" campaign. You'd have a "WTO" campaign, which wouldn't be able to try and reassure people that nothing would really change (As numerous Leave backers did). And a just-like-Norway campaign. And a just-like-Switzerland campaign. And a just-like-Canada campaign. And a full-membership-of-the-EU campaign. Each having to actually discuss the different approaches, and why their one was better than the others.
And you'd, hopefully, get an idea not just of "Should we leave or not", but also "Where should we go?". And a more nuanced discussion of costs and benefits than "I hate them!" vs "You're fascists!"
Now, I fully concede that the government might not be able to then achieve what the public asked for. They might _want_ to join the EEA, but be told to sod off by Norway. But then they'd have to try and get as close as possible to that, rather than saying "We got a split vote and a country which is pulling in two different directions, so we're going for the most extreme Brexit possible".
(Which isn't what most people want - actual British attitudes are more nuanced than that.)
Of course, I _don't_ run the country. And instead the person who does is concentrating on not splitting the Conservative Party rather than anything actually useful.
Original post on Dreamwidth - there are comments there.