Many "Remainers" are writing to their MPs, emphasising the referendum was "advisory" and sometimes demanding a second referendum. I think both of those are damaging and alienating ideas to cling to, they won't help to fix our politics.
The ideal way forward is for the next PM to get some informal outline, the likely shape of a UK/EU deal, and put it to Parliament at the same time as a bill authorising them to trigger Article 50. We know that the Leave campaign's fairytale deal is not going to be on the table. So, armed with better information, Parliament could then choose to enable or refuse the triggering of Article 50. I don't know what the outcome would be; but it would be the right way for our democracy to work, and it leaves open a route "back into the EU" now that some of the referendum's consequences (not least, for the future of the UK as a union) are crystal clear.