According to a survey on the BBC History magazine, the anniversary of the signing of the Magna Carta is the best day to celebrate Britishness .
Personally I would prefer some event after the founding of the United Kingdom, such as the abolition of the Slave trade, but the signing of the Magna Carta isn't just an event in a school textbook. It marked the start of our unwritten constitution. You can put the survival of the monarchy and our democratic traditions down to this, as not only did it show English monarchs that they could not rule carte blanche over the nation, but that it shows the British resilience to adapt to the changing movements of our times, before those movements imploded. Hence why we managed to keep our monarchy and France, Russia and various other nations did not.