Well, what will happen in the UK in the next couple of weeks/months/years might just be enough to show by example what the good bits of being a part of the EU are.
The big mistake the brexit voters made is that they thought that the UK as it was before the EU would have been a better place to be in today if the EU had never happened. Contrary to their belief the sum is a lot more than the parts but explaining that to the electorate is a challenge that modern politics (soundbites and tweets) is badly equipped to deal with.
Much easier to push the emotional buttons of xenophobia and economic malaise due to 'others'.
Today's EU is like America prior to the ratification of the US constitution. It's not suitable for long-term functionality.
You probably ought to convince Switzerland (an enclave) and Norway to join. You probably ought to convince Greece to leave. You then need federal taxes, a unified military command, a head of state, a single EU citizenship, and all the other trappings of a real country.
As a fellow European I thought about this too, as I was trying to see a silver lining in all this. And then I went to read the comments on the French paper's Le Figaro website and got depressed again. Most of the people commenting in there really believed that the EU is to be accused of everything that's been going wrong with the French economy. If we do get a French referendum and if France leaves then the EU will just die. It pains me to put this down in words, but it's a valid possibility.