Hi Gums,
Have you tried the website
www.historicaldirectories.org they have many searchable directories on line including the Post Office Directory of Northumberland & Durham 1858. If you do go onto the site I would suggest that you spend a few minutes exploring how to get the best out of the site first.
There are several Bell families who ran pubs in County Durham and the Durham Publicans start on page 652 of the directory mentioned above. Not sure that the images and pages have the same numbers - you would need to check. In the directory the list is alphabetical by pubs but these are a few of those I noticed ran by people called Bell. There could be others!
Angel - R. Bell, Market Place, Hartlepool - page 652
Beehive - T. Bell, Low Fell, Gateshead - page 652
Black & Gray - J. Bell, 21 Union Alley, South Shields - page 652
Blue Bell - G. Bell, Middleton-in-Tees Dale, Barnard Castle - page 653
Clarence - W. Bell, Commercial Street, Stockton - page 654
Fulwell Inn - Mrs. E. Bell, Fulwell, Sunderland - page 656
Hare & Hounds - Mrs. D. Bell, Bishop Middleham, Durham - page 657
Hawk's Arms - J. Bell, 32 East Street, Gateshead - page 657
Holy Island Castle - R. Bell, 20 Coronation Street, Sunderland - page 657
Napoleon - J. Bell, 36 George Street, Sunderland - page 658
Mary Queen of Scotts - J. Bell, Heworth Place, High Street, Gateshead - page 658
Newcastle Arms - R. Bell, East Hetton, Ferry Hill - page 659
Queens Head - T. Bell, Crook, Durham - page 659
Roker Hotel - J. Bell, 9, 10 & 11 Terrace, Rocker M. Sunderland - page 660
Squire's Arms - W. Bell, Heworth Shore, Lower Heworth, Gateshead - page 661
Pelican - John Bell, 23 Sunderland Street, B. Sunderland - page 659
White Lion, John Bell, Stanhope, Darlington - page 663
I hope this might be of some help, and I can definately recommend a look at the Historical Directories website.
Good Searching,
Patsy