I have done some research and all the cheap backpacker type hotels are located in the Sultanahmet district which is located between the Blue Mosque and the sea.
Check out google maps and the backpackers hotel websites.
When are you going to Turkey? I am leaving on 1st June from Dover.
GR
You may want to check this one out too; http://www.hotelangelshome.com/
Nice, cozy place right in the middle of old city in Sultanahmet region. The owner/manager is a biker too, he even has the banner of his bike club in the bottom of the hotel's website. Some local bike clubs even have their meets in the hotel's cafe next door.
I guess he will be more than happy to help you out if you call ahead and say you're coming down all the way from U.K on a bike.