Hi all
David, Durham Place was off Durham Street in Middlesbrough. There was also a Hilda Place off Durham Street, too. They were both between East Street and Commercial Street, a bit further along the road from your suggestion of between Queens Square and the Captain Cook pub. They were both cul-de-sacs (the posh word) or dead ends (the not so posh word!) I drive down Commercial and Durham Streets when I go to work - but there are unfortunately very few original buildings left in this area of Middlesbrough.
I don't have personal knowledge of Durham Place itself, but being very interested in the history of Middlesbrough, have known its location for a while now. It appears on a 1913 map of Middlesbrough which you could buy very cheaply from the Alan Godfrey ordnance survey map site. It's the North Middlesbrough (St Hilda's and Port Clarence) 1913 map you need.
There is even a very good photograph of Durham Place in Araf Chohan's book, Middlesbrough, which is part of the Britain in Old Photographs series. I don't think I'd be allowed to post a scan of the photo here unfortunately. But maybe you could get the book through your library? The ISBN is 1 84015 1404. I'm sure you'd be fascinated to see it if your gg grandmother was born there! The book says it was off Cleveland Street, but Cleveland Street changes to Durham Street once you get past the East Street junction.
In the undated photo it looks to be a poor, narrow street of two rows of terraced houses, flat-fronted, with a stand pipe in the middle of the cobbled road. No pavements as such, and the book says there was a pub, the Golden Lion, on the left corner which is just visible in the photo. A brick wall with some sort of door or window - maybe blocked up, I can't quite make it out - or it may even be an advertising board, can be seen at the end of the street, and I can see a very faint building behind the wall - looks like a roof with chimneys. But when I look at the 1913 map, the old St Hilda's graveyard should have been behind this wall, so perhaps the photo is later than the map, and more buildings have been erected. Or maybe the building is a long way in the distance. There are a few people in the photo, children sitting on the cobbles, a couple of people in doorways, etc. The women are wearing long clothes, ankle length, possibly covered in aprons. It looks to me to be from around the 1910s or something like that.
Sorry for the long post, but I hope you find this information interesting.
whipby