Also look at TA forums for both towns.
Many thanks for the recommendation.
Hi tfish; thanks for saying thanks; not that many do!
Using the motorway routes and following an odd but deceptively quick route from London to Birmingham to Carlisle to Glasgow to Edinburgh allows you to stop at the Lake District, Lancaster, Liverpool ... (it looks strange but it really is as quick as the Eastern route ... traffic allowing ... I've driven it several times).
You wouldn't necessarily want to stay the night in all of these places but they all offer something different with regard to history and architecture.
It's also worth mentioning that driving from London to Edinburgh, with no traffic issues, on the motorway will take approx 6 hours (plus rest stops).
Hi tfish; other good ideas coming in; Chester is a town not to be missed if you're over on the west coast.
I read your question as basically a one night stop and one day visit on route and that's what I answered. If it is that, I very strongly suggest the train as both York and Durham are major and very easy stops by train. Both are a little bit off the main road routes north. Edinburgh rail station is right in the centre of the city.
Hiring in London could bring the heavy congestion charge, will bring driving through heavy congestion, etc. Whatever road route you use north of London, you will be in heavy traffic for many many miles. Unless you are well out of central Edinburgh, there is next to no parking in the city and the wardens are even more fearsome than London's. Driving is bad also and some main streets will be closed due to a new tram (streetcar) network being built. Public transport is excellent in Edinburgh and most sites; old town, castle, museums, art galleries, and much more are walking distance or a short local "hop" on a bus. Hire your vehicle as you leave the city (if you want automatic, you have to specify as we are still into stick shifts here!)
If you do have more time please say and we can add many many towns to the list.
PS; if you are considering having a car in Edinburgh; DO look at the Edinburgh forum. The Edinburgh locals will soon convince you not to !! It's a stunning city to walk round. It's a stunningly bad city to drive or park in!
If you decide to drive and take the East side of the country you could take in Lindisfarne (as someone already mentioned) as well as Bamburgh Castle and Alnwick Castle. Alnwick has an amazing water gardens as well as being a movie location for a number of the Harry Potter Films.
York would be an excellent stop especially if you go by train. Parking can be a bit difficult in York.