There is a variety of accommodation available for a family staying for just a short break. My overall choice for such a trip would be to stay in a bed and breakfast for the duration because it is charged nightly and whilst it may cost more, breakfast is always included as part of the price.
One place that comes highly recommended with a 3 star rating is the Friars Rest Guest House, it is located on Fulford Road, just a 5 minute walk to the centre, the guest house is suitable for families and prices start from a modest £19 per night, per person. It is good value for money for just a weekend stay. For further information or bookings, you can contact the Friars Rest on (01904) 629823.