You and I may hold the opinion it was based on a lie but that's not how you challenge a democratic decision. Who is going to determine whether the referendum was fairly campaigned on or organised? Which UK Court is going to accept a challenge? What's the legal basis for doing so?
There is none.
I don't think the referendum should be re-run. Nor should it, the majority of people voted to leave.
I think were things get thorny though is the Northern Ireland and particularly Scottish issue. I know for many English voters the SNP are seen the northern faction of the PLO but once you look past bias nothing they are saying is unreasonable. A huge majority of their country voted to remain and every one of their 32 constituencies voted to remain, yet they face themselves being dragged out of Europe. Whether a UK court would accept that as being undemocratic I don't know but a European court would look at that very closely. When you study the Lisbon Treaty article 50 I find it very hard to reconcile the terms being met when you consider the situation with Scotland. If the EU allowed it to go ahead it would make a mockery of their legal system (especially when they rejected the Greek referendum on a similar basis 2 years ago as not meeting the standards required).
People won't like how the SNP are behaving but all they are doing is trying to reflect their own countries vote (which was a 600% higher majority for remain when compared to that of leave across the 4 countries). There is a sharp contrast but in all fairness that is not the SNP's or Scotland's doing, the fault line lies firmly with the government who have not considered how divisive this could be.
Much of the legalistic discussion will revolve around how far can Scotland be considered a country. It has it's own Parliament, sets increasing rules on taxation, education health and fiscal currency as well as having it's own government. Whether that stands up or not is up for debate but I would urge anyone to read the article 50 of the Lisbon treaty and ask, if they lived in a country where every one of the 32 states voted to remain whether it would be democratic to drag said country out.