Frankly let them be lynched. I would love a rebound leader willing to instantly fall on their sword by doing something to actively stop this. Would actually give me a tiny smidge of respect for that party. But probably what we'll get is political infighting and back room deals.
At minimum I think calling an election would be a start. Allows parties to run on a 'fuck the referendudm' ticket if they want.
Talk to the voters about why they really voted leave. If it was a protest look at the underlying issues. Immigration is clearly one, but is there anything that can be done to reasonably tighten things up around requiring a job before movement of people? Separate the issue of internal EU migration and external - the Syria crisis will have focused attention on immigration again and maybe the two issues are being conflated in some minds.
Talk about how much bullshit the newspapers talk about the EU pulling the strings and Westminster having no control. About how Europe provides investment for poorer regions, protects their workers rights from being eroded, how some of our biggest manufacturing plants come from inward investment which relies on having access to the European market.
Get fucking Martin Lewis on to talk about it - people will probably listen to him more than politicians
People needed to be talked down from the ledge during the referendum campaign but they weren't - they were shouted at. "If you jump it'll fucking hurt you twat" rather than "it doesn't need to be like this, we can make it better"