I wasn't proposing an hourly service, rather that a substantially improved timetable, over the current one, is entirely possible through 'timetabling measures' rather than 'infrastructure measures' such as new curves, re-instated loops, sections of double track etc.
The summer 1983 timetable was essentially two-hourly with some gaps filled. Ex Middlesbrough 0938, 1042 and 1142 giving a range of services for daytrippers rather than the current 'everyone onto a single train at 1020'. Ex Whitby at 0653, 0910, 1110, 1315, 1522, 1610, 1715, 1820 and 1925. Nicely providing a good range of departures from Whitby to meet day-tripper needs.
As has been suggested, the most immediate and easiest solution is to strengthen the existing 1020 Middlesbrough departure. Perhaps utilising the unit which arrives Middlebrough ECS ex Nunthorpe at 1002 and has no apparent work for the remainder of the day.
Beyond that, timetable enhancements in marginal time, such as the proposed commuter service (which does not require additional rolling stock). Similarly a later evening service, as is provided on summer Fridays.
As an awful suggestion, Arriva now seem to get a decent bus up the valley, road vehicles having moved on since the 1960's. Perhaps the schoolkids could now go by bus instead and the line be closed! Or handed over to preservation!