Northern Ireland has its own distinct culture
Proponents for an independent Northern Ireland have surfaced on and off since the Partition. Such fringe political parties include the "Ulster Third Way" or the "Ulster Independence Movement" and appeal to Ulster nationalism, Ulster identity, a common Northern Ireland identity, and peace in Northern Ireland. None of these political parties have gained traction since the "Ulster Third Way" party was deregistered in December 2005.
Due to Northern Ireland's unique relationship with Britain, the citizens of this region have adopted a culture that remains a rare blend of Britishness and Irishness. This is specifically evident in the presence of both nationalists (those who identify as Irish and are primarily Catholic), and unionists (those who identify as British and are mostly Protestant). However, due to such a mixture, Northern Ireland fails to fit into the cultures of both Britain and the Republic of Ireland. Thus, as a result, both Irish reunification and maintaining a position in the United Kingdom will fail to solve the problems faced by Northern Ireland— especially in the wake of Brexit. Some of which include extremely high poverty and unemployment rates.  Moreover, remaining in the United Kingdom or reunifying with the Republic of Ireland entails the issues faced by Northern Ireland to be repeatedly overlooked. As a result, Northern Ireland fails to be represented. As previously mentioned, differences regarding Brexit have exacerbated a "culture war" between Britishness and Irishness— a presence that has existed far before the debate came to a head. Specifically— beginning in the mid-1990s— as Northern Ireland moved toward a peace settlement, a series of disputes between nationalists and unionists led to violent street clashes that almost derailed the process entirely. Similar events transpired in the early 2000s, a few years after the signing of the Good Friday Agreement. Brexit brings these dormant tensions back to a boil when considering cultural implications. In standing with the rest of Britain and, in turn, Brexit, allows only the aspirations of the unionists to be met. Therefore, it is in Northern Ireland's best interest to become an independent EU state and as a result, become its own priority. 
The differences between the cultures of Northern Ireland and England appear only in their names. In other words, Northern Ireland follows closely in England's cultural footsteps. Therefore, regardless of the implications of Brexit, Northern Ireland remains extremely similar to the rest of Britain in regards to language, recreational activities, popular food items, social behaviors, and in some regions of Northern Ireland: religion. As a result, since Northern Ireland is too financially ill-equipped to manage their own independence, citizens of Northern Ireland will simply be happier in unification with a country that reflects their own values. The Republic of Ireland, on the other hand, actively tries to differentiate themselves from the rest of the United Kingdom— and as a result, they distance themselves from Northern Ireland. Thus, as Northern Ireland's culture remains closer to that of their British counterparts, they should remain within the United Kingdom, where they belong.