Yes, courts will overturn the Brexit vote. Enough uninformed voters who chose to support Brexit are already experiencing cases of buyer's remorse due to the potential repercussions this decision would have on the lives of all Brits. The courts have a responsibility to keep the best interests of these people in mind.
I'm not really in the know on all the details of Brexit, but I do know that it sounds more and more like it wasn't such a great idea after all. It seems that many people voted for it without knowing how much harm it could really cause. If that is true, then it is very possible that courts could overturn it to save their country.
The people knew what they were voting for when they voted for Brexit. The people knew that they wanted out of the EU and they voted for it out of their own free will. The courts do not have enough power to overturn what the people clearly voted for as a popular vote.
I find it unlikely that courts will overturn Brexit. So many in the UK are adamant about it and have no desire to return the EU. Perhaps if the negative affects become so great then the public will reconsider and the courts will then rule against it. But I believe that short term there will be no backing out.