(2-part post):
As Mushroom25 states, there are several online car hire brokers who (just like insurance brokers) will all claim that they'll get you the cheapest price. However, my preferred option would be to pick up the telephone and ring round lots of the firms in your area. (Don't forget to mention that you'll be taking the car abroad). Then you'll be be able to discuss the various options, such as the type of vehicle you require. Just look in Yellow Pages under 'Car Hire - Self Drive'.
However, you asked for an estimate of cost. Based on a quick look at
http://www.holidayautos.co.uk , prices start from around �180 for a Ford Ka or �230 for a Ford Mondeo.
As I assume that this is a follow-up to your previous question, I'll tell you that ViaMichelin estimates the total driving time, from Newcastle to Barcelona, as 18 hours 55 minutes. That excludes the time waiting at Dover for the ferry and off-loading in Calais. It also excludes any breaks. Realistically, you're looking at two full days travel, with two people sharing the driving. e.g. leave early on a Saturday morning and share the driving until the early evening. Stop overnight in France and then spend another day driving, before arriving in Barcelona (or on the Costa Brava) on Sunday evening.
ViaMichelin also estimates the total (one way) cost of the journey as approx �250 (about �170), with two thirds of the cost being petrol and one third being tolls. (Remember that you'd also have to pay for the ferry and for overnight accommodation).