In fact, it can be proved that Hamiltonian Cycle remains $\mathsf{NP}$-hard even for a very restricted subclass of line graphs: line graphs of $1$-subdivisions of planar cubic bipartite graphs. These graphs are cubic. Not surprisingly the same holds if we pass to $4$-regular line graphs: Hamiltonian Cycle remains $\mathsf{NP}$-hard for line graphs of planar cubic bipartite graphs.

Both results can be obtained by reductions from Hamiltonian Cycle restricted to planar cubic bipartite graphs:

T. Akiyama, T. Nishizeki, N. Saito, NP-Completeness of the Hamiltonian Cycle Problem for Bipartite Graphs, Journal of
Information Processing 3 (1980) 73-76.