I'll knock on wood to be safe, but a pre-E3 Direct in April or May seems inevitable... though myself and others have said such things before about their timing and been months off.
Hopefully said Direct will get near future things out of the way like ARMS release date/overview, known 3DS releases (Ever Oasis, Pikmin, Bye-Bye BoxBoy!, Lady Layton), and various amiibo announcements for released and near-future games. Also I hope said Direct will at least announce and give basic details on Virtual Console. (That said, certain hotly demanded VC announcements like GameCube games and Mother 3 would be quite appropriate cake icing at E3).
My conservative predictions for E3 itself. Obviously the soon-to-be-released Splatoon 2/online service details, Super Mario Odyssey, Xenoblade Chronicles 2, FE Warriors, and at least a peek at the next mainline FE. Probably looks at Sonic and Skyrim and one more third party surprise. Indies may or may not have their own mini Direct around this time. There is probably one big unannounced 2017 first party game (Smash 4 port is where my money is), and we will likely get 1 or 2 big 2018 game announcements (Animal Crossing, Retro Studios project, Pikmin?)
My super dark horse, wild card prediction is Eternal Darkness 2.