B777 Captain reported encountering wake turbulence on initial climb out of HNL from another B777 landing on an intersecting runway.

Narrative

I was acting as Check Airman in the right seat training a Captain new to the B-777. After holding short of Runway 08R; we were given clearance to takeoff on Runway 8R with a heading of 155 degrees after departure. In addition; we were asked to expedite due to traffic turning final for Runway 04R. Not wanting to rush my student I just repeated the cleared for takeoff clearance and completed the before takeoff checklist. Prior to moving the throttles forward; HNL Tower canceled our takeoff clearance while we were in position. After the B777 cleared overhead our runway centerline; we were once again cleared for takeoff. We took off on Runway 8R. On departure we flew through the wake turbulence of the prior aircraft which caused my student to initially think he might have had a tire blown. I mentioned it was wake turbulence and told him to continue. No further issues noted. Trying to time departures and arrivals for intersecting runways. Not enough time allowed to taxi a big wide body aircraft into position. Also; no allowing for the wake turbulence of the landing aircraft on the departing aircraft's takeoff roll. Allow more time for wide body aircraft.