(More generally, I'm curious. Has there ever been a successful petition on the government website or change.org? I.e. one where the government said "Yes, you're right, we'll do the thing that you're asking for." I haven't heard of any.)
Taking a specific example, there was petition 223729 in July: Rescind Art.50 if Vote Leave has broken Electoral Laws regarding 2016 referendum. This got more than 100,000 signatures, so Parliament debated it in September. The debate lasted 1h09m: you can watch the video or read a transcript via that link. It's not quite what I expected: for one thing, it didn't take place in the main chamber. Instead, it was in a smaller committee room. By my count, there were fewer than 20 people present, and none of them were party leaders. It was also far more civilised than any of the debates I've seen/heard in the chamber, where people are booing each other as if they were at a football match; I can only stand a couple of minutes of that before I'm filled with rage. In this case, there were MPs from different parties (Labour, LibDem, Conservative, SNP) all working together. They were all pro-Remain except for The Parliamentary Under-Secretary of State for Exiting the European Union (Chris Heaton-Harris), who gave a speech at the end that trotted out the standard party line: "blah blah will of the people blah blah" (paraphrased).
In brief, we know that the Leave campaign broke the law, and they were fined by the Electoral Commission. If the referendum had been legally binding (like a general election) then this would be grounds to discard the results and start again. However, since the referendum was advisory, the government are free to follow or ignore the results as they see fit. That means that there's no legal basis to demand a new referendum. The frustrating point is that the government keep claiming that they are bound by the results.
I think the wider issue is that our current laws don't handle referendums very well, possibly because they're so rare. Looking back at the AV referendum from 2011, the "No" campaign lied about the cost (£250 million); for instance, they included the cost of the referendum itself (a sunk cost) and assumed that we'd switch to electronic voting machines. As I wrote at the time (Referendum results), the campaigners are legally allowed to lie, as long as they don't lie about rival candidates. In the Brexit campaign, the £350 million printed on the side of the bus was also a lie, but that's not why the Leave campaign got into trouble; they only broke the law about overspending.
So, suppose that there was a new referendum. How can we trust that this would be any more honest than the last two? Look at how brazen Donald Trump has been in the USA, denying that he said things even when there's video evidence to the contrary. Could we see the same thing over here? More to the point, can we assume that the electorate will actually know (or care) what's true?
Being charitable, let's say that all the Leave voters saw through the lies, but still decided that we'd be better off outside the EU. That doesn't mean that they all had the same vision of what leaving would involve, e.g. how to handle the Irish border. As I said before (Unintended consequences), I would ideally like to see a multi-way ballot based on AV. Some people might say "Any kind of Brexit is better than staying in" while other people might say "Keeping our current status is better than leaving the EU and joining the Schengen Zone".
Suppose that there was another general election under the current party leadership: as a Remainer, who should I vote for?
- The Conservatives are definitely out, since Theresa May is the one pushing Brexit!
- Labour are also out, since Jeremy Corbyn is also pro-Brexit. (I don't think he's officially supported it, but he's been pretty luke-warm in his opposition.)
- The Liberal Democrats say that they're against it; I've been sceptical of them since they went back on their promise about tuition fees, but I decided that it was time to give them a second chance. Then there was a vote where Vince Cable (current leader) and Tim Farron (former leader) didn't bother to turn up, and the government's amendment passed by 3 votes. So, they're out too.
- I think that only leaves the Green party who are willing to make Remain a party policy.
Ultimately, some of the damage has already been done, e.g. the European Medicines Agency (EMA) are moving their head office from London to Amsterdam. However, I think we can and should stop things from getting too much worse.