Originally the Rowntree (Quaker confectioners) mansion, this handsome Victorian house makes a spacious and child-friendly youth hostel, with more than 250 beds, a broad garden and on-site restaurant. It's often busy, so book early. It's about a mile northwest of the city centre; there's a riverside footpath from Lendal Bridge (poorly lit, so avoid after dark).
Alternatively, take bus 2 from the train station or Museum St, though be aware it doesn't run in the evenings.