Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Waikiki to Oslo as you normally would.
Due to multiple factors, expect to potentially pay more when booking in July. For low rates, September can be a great choice for travel to Oslo. Flights to Oslo from Waikiki in September are about $1,162 on average during September, but can be found for as low as $541.