Locations that might be good:
Cherry Red’s (there are a couple of meetups here already; in the city centre)
The Victoria (ditto, ditto)
York’s Coffee (not a pub, so more open to people who don’t drink or are under 18 or just don’t like pubs)
There are also quite a few gaming cafes around; Geek Retreat and Manaleak, and they may have room (I’m not sure). I’m pretty sure there used to be a board gaming thing in Urban Coffee as well. Afternoon Play, who are a board game group, are on Sundays in Revolution on Broad Street, so they may know some more venue ideas.