Hard problems on subclasses of planar cubic bipartite graphs
Hamiltonicity remains NP-complete for 2-vertex-connected cubic planar bipartite graphs, but this is not actually a restriction: every cubic planar bipartite graph is 2-connected. It is a well-known open problem whether 3-connected cubic planar bipartite graphs always have Hamiltonian cycles, but if they do not then the problem is NP-complete for them as well.

(One of my recent papers looked at some related connectivity classes of cubic planar bipartite graphs but not from the point of view of NP-hardness.)