That could just be because of the avaliable paths. i.e if it stops there it could hold up trains.
The service is either timetabled to stop there or it isn't. You can't randomly pick and choose on a day to day basis based on pathing, and in any case that certainly would not just be done on the driver's initiative. It would only stop there if there is either information on the driver's diagram (which may cover a service alteration for the day), a special stop order to cover a cancelled service, the signaller says so (as above), or some other deliberate instruction. Otherwise, if it isn't in the timetable, it may stop at a red signal but the doors shouldn't be opened.
Bear in mind that many trains will not be travelling at significant speed on the Slow lines at that time of the morning anyway - there's not a whole lot of flexibility between Balham and Falcon Junction to make up time if something is so much as fractionally late. So it is a bit nonsensical to further complicate matters by randomly choosing whether or not to stop. It is evident that, at that particular time, it has been deemed neither necessary nor efficient to usually stop that train at Wandsworth Common. So it shouldn't, except as above.
If the train is advertised incorrectly, this is most likely due to the driver having been supplied with an incorrect PIS code (the code entered into the passenger information system which tells it the route and calling pattern). Unfortunately the ability for the driver to change this will depend very much on the stock, the time available when stationary to fiddle around changing the system, and so on. A manual announcement is probably going to suffice on most occasions. It's not actually hard to get back to Wandsworth Common if you do miss the stop, anyway. And if the train
is ordered to stop, it doesn't need any alterations to the PIS anyway, so everyone's a winner then.