If they vote by a small margin to stay, then does than mean we have another referendum? What happens if this then says leave.
Three million signatures. There were more than 3 million who voted to leave so the margin of winning is irrelevant as a litmus test.
There not only is no need for another referendum; there was no need for the first one, which inevitably was going to cause more harm than good regardless of the result.
It isn't legally binding on the government, and there are a large number of scenarios whereby article 50 might never be triggered - particularly given that Cameron has announced his resignation, but won't go immediately.
Whoever replaces him will do so long after the public have moved on to the next big excitement - perhaps the Rio Olympics - and there need not be much pressure on the new PM to act - there's no legal requirement for him to do so, and there are any number of ways to keep stalling on the issue.
The vote was close enough that any politician who nails his colours to one or other mast stands to lose as much as he gains (with the notable exception of UKIP, which is a single issue party).
If opposing the triggering of article 50 wins the votes of almost half of the country (the half that voted 'Remain'), and if a commitment to immediate triggering wins only the non-UKIP 'Leave' voters, then a completely cynical analysis says that the smart move for both Labour and Conservative is to be as slow and obstructing as possible, with a view to quietly shelving the article 50 trigger at some future date.
The side that backs Brexit gets to share half of the votes with the kippers; while their opponents get the other half all to themselves. The game theory optimum strategy is not to commit to Brexit, but to try to persuade your opponent to back themselves into that corner. Coming out against Brexit is a winning strategy, but only if done long enough after the referendum to avoid a massive backlash that could lead to UKIP becoming a major player.