Ah, the weekend of the Great North Run.... good luck! Newcastle will have been booked for ages. If you can't get anywhere in Gateshead either, try Sunderland, only 25mins on the metro and probably quieter that weekend.
Just done a search and there is the Express by HI Newcastle metro which is at swalwell which is only a few miles away that has availability it came up on roomsandrates.com ther was also a hotel at boldon which is between sunderland & newcastle but everywhere is pretty much booked up, always is at that time of year
on the other hand there is a caravan park in Whitley Bay