This is a great idea. Are you going to re-name the pub as well? How about The Cyclist's Legs?
Seriously though, this is what I look for in a stopping place:
Firstly: somewhere I can sit and keep an eye on the bike because I don't carry a lock. The best place is an enclosed patio or garden from where it would be difficult for a thief to remove the bikes to a waiting van unseen.
Plenty of hot tea or coffee - the Clarion Cafe in Pendle supplies tea at 50p a pint and you could do the American thing with unlimited coffee topups because most cyclists wouldn't want more than one mug anyway.
A selection of cakes and light snacks. Cyclists don't want a full meal while out on the road.
Water, a pump, magazines, maybe sell a selection of spares like inner tubes and gloves, have a look at the website of the Green Jersey bike shop in Clitheroe.
As well as this, cyclists want somewhere warm and cosy to relax and chat with pals so a quiet, warm, well-ventilated room is good. Puddleducks cafe in Dunsop Bridge, the centre of the UK, used to be a terribly steamy place in winter with uncomfortable humidity and condensation running down the windows until the owners installed a woodburning stove, which warms the room while sucking massive volumes of air up the flue. It is now a much more pleasant place. No music, no TV and chairs that don't scrape on the floor. Oh, and nice clean loos with space for washing and changing.
The Alpine hut shoe suggestion is quite a good one but I suspect the flip-flops would all get pinched.