Yes OK point taken, but if you could type in a starting station and show 'fill em up' bargains for the following day I am sure people would start to use it, if a genuine saving can be made people will go for it, and maybe do a journey which they wouldn't otherwise be able to/wish to afford
I would absolutely use such a service, no question of it. It's ideal for those of us who want a day out but not super fussed about where and have flexibility. I strongly suspect even families could use it.
As for whoever said no-one plans their days out the same day/the night before, oh that's so not true. My day out yesterday was being mostly planned on Tuesday morning, but not booked until something silly like 2330 Tuesday night, with the travel beginning at 0857 on Wednesday. I knew where I was going, or at least I had various options, so I looked them all up before committing to the adventure.
I regularly don't buy tickets to travel until the day itself, sometimes with less than an hour's notice. mike57's idea would give me that flexibility to go "Ooh, I didn't think of going there, I'll book that" or "Oh wow, that journey's a bargain today, I'll book that one and not the usual one". Of course, fares being set and not flexing on demand means it is more reliably known in advance how much a journey will cost, which can be very useful. Especially on regular journeys, where it can all be booked on the walk to the station and it being known what time that fare is valid at for the return.
Regarding weekend fares, I'd be against putting them up too. It wouldn't directly affect me, as I have extremely low to zero need of the railway at weekends, but for the benefit of humanity as a whole I don't think it's a good idea.
Talking of better for humanity, I'd instead look to actively enforce no smoking outside entrances to stations, including paths directly leading to the entrances. Vaping would be included in that, and equally applicable to railway staff and passengers, and of course everyone else, alike. It's certainly one thing that would help to encourage people to use the railway, and yes I include myself in being frustrated by smokers at stations as it's not fun having to walk past armies of smokers and vapers to get onto/off the stations. I would honestly say that and providing a service which is reliable is more important than jacking up rail fares.
Quite honestly, the prices asked for a lot of journeys puts me off using the train. I'm not the only person I know who has that issue, I know people who have railcards who even with the discount don't use the trains because of the cost and unreliability. If the fares were to go up at the weekends, then as so many others have already said, just you watch the demand levels drop and the roads get much busier. It may not happen overnight, but by the end of the year it would be noticeable.
If it's the peak time trains that aren't loaded to the maximum any more, but leisure time trains are, then the solution for me would be to scrap peak time altogether. That would sure spread the loads out during the day, and I would be much happier to use the railway at say 7am rather than wait until say half 9 for off-peak to start. Making the railway affordable for the masses has to be the way forward, there is zero point running a 12 car 700 at an average fare of (say) £30 but it being lightly loaded when you could sell the same train at an average of £15 and fill all 12 coaches. It is always better to have 100x £15 than maybe 30x £30, especially if the route in question has on-board catering, the lower fare means there's a chance the average person would have more money to spend on a £3 coffee and snack deal from the trolley for example.
So, weekend fares increase? No thank you. Scrapping peak time and increasing the amount of people on board, even if the average fare is less, yes please.
