Buying 'Advance' tickets will be the cheapest way to do this journey - these are the tickets that restrict you to one train, but are much cheaper.
If you are able to travel in the morning, you can currently do the journey more cheaply: there are currently fares of £24.60 on the 09:30, 10:30 and 11:30 trains to Durham, and £18.45 on the 08:30. Otherwise, the above quote of about £31 is the best for travel in the afternoon. As the number of tickets at each price brand is limited, it is likely that they will at some point sell out. The tickets for the return leg are likely to be a similar price, when they become available. Tickets are usually released once a week, on a Friday. You can buy the outward and return journeys separately if you want, so could get the outward ticket now.
The £80 Super Off-Peak Return is valid after 09:30 on the Friday and all day Sunday - it would give you more flexibility. You can still book seats with it, but you aren't tied to having to use that train.
As daniel78 says, depending on your local station, you may be better buying a through ticket - which station would you be travelling from?
As a final note: if you're really needing to save, you can sometimes shave off a few pounds by 'splitting' your journey at Newark. For example, an Advance fare on the 14:08 London-Newark is currently £6.65, getting in at 15:34. You can then pick up the 15:43 from Newark-Durham for £18.45. Again, these are Advance tickets so they may sell out and relative price savings against a through ticket vary during the day. Similar savings can sometimes be made by splitting at Doncaster. If you're interested in doing this, it's worth playing around with the journey planner. As long as you leave the Minimum Connection Time (5 minutes at Newark, 7 minutes at Doncaster) between your train arriving and the next one leaving, then your tickets are still valid in the event of delays.
