Join the Schengen Agreement
The UK and Ireland are the only EU countries not committed to joining the Schengen Agreement. The Agreement is working well in almost all of the EU and there is no particular reason that the UK should not be a member. Why does Britain being an island make any difference? We can simply allow anybody in from other members of the agreement like most of the EU does without any trouble. This would simply put our liberties on a par with other EU countries, so it is hardly a huge and radical step. But it would make travelling much simpler.
Why does this idea matter?
It would make this country on the same level of freedom in this regard with the bulk of the EU and would increase the integration of the UK with the rest of Europe. It would mean that the whole of the EU functioned like one country, with freedom of travel. Freedom of travel is one of the fundamental human rights.