I suppose they could have something like a 'Stop' button on the platform where you simply press the button if you want the train to stop, and something like 'PAX' lights up on the signal before the station, similar to an Off Indicator. If that makes sense. Obviously that isn't very helpful on branches with no little to no signalling.
Anyway I rather think Cheddington during the off-peak hours should be a request stop. Perhaps Longcross too. I always thought Brundall Gardens was a request stop but my trains through there in my last Anglia visit both stopped there and nobody got on or off either time.
Longcross just wouldn't work as a request stop. The services through there are just too busy and too "important". Now, if we use a different approach, we could extend the Camberley services from Ascot, and have hourly calls at Longcross. The main reason Longcross performs poorly is because SWT constantly change the calling pattern. The 1242 from Reading and the 1250 from Waterloo stop there; otherwise, it changes with every timetable!
Also, I notice that no-one has flagged my Battlesbridge suggestion. Having never been on the line through Battlesbridge, I can't tell you how busy each journey is, but it has only a few thousand travellers each year and something like 40 journeys a day. On the flip side, I have been over the (narrow) bridge there enough to know that it is quiet. I have never seen a passenger there in my life. Mind you, I would have thought that an hourly bus to Southend an Chelmsford would suffice... if it still existed.
As for the "stop" idea, they use a similar system on some Metros, which works really well. I first encountered this idea in Bonn in May '03, and thought that it was completely ridiculous. But each time the train would stop at a station, find the platform deserted, and then accelerate immediately. The obvious issue of "why not merely slow to a crawl" aside, the idea never seemed to fail whenever I travelled on it. And the 66 route (don't know if that number still applies; it linked Bonn to Siegburg for the Cologne-Frankfurt ICE route) was always very busy. Yet I distinctly remember one station from the southern end being a minor halt, only a few hundred metres from the terminus. Such a station was convenient - and easy to skip without any trouble. Back on course, the driver could be equipped with a mobile device that could switch off the "Stop" sign on the platform, just like a remote.
As for us lot, we could always erect a large sign with big red letters saying,
DO NOT STOP HERE. WE DO NOT INTEND TO TRAVEL AT THIS TIME.