What would this aromatic boron compound be called?
According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), heteromonocyclic compounds with up to and including ten ring members are named by the extended Hantzsch–Widman system. According to P-22.214.171.124.2, this also applies to heterocyclic parent hydrides containing Group 13 (boron group) atoms.
The Hantzsch-Widman system stem for an unsatured 7-membered ring is “epine”. Therefore, the preferred name for the compound that is given in the question is 1H-borepine.
The name 1H-1-boracyclohepta-2,4,6-triene is not recommended.