Train from London to Edinburgh with a stop in York

Depends what sort of ticket it is. The cheapest tickest are only valid on the train listed on the ticket. More expensive tickest are valid on any train.
Best way is to buy two cheap tickets, London to York and York to Edinburgh, that will be cheaper than one full price ticket London to Edinburgh.

Is it a good idea - NO. York deserves more than a few hours. Stay overnight.

Left luggage facilities aren't usual at UK stations now since the IRA terrorism of the 70s/80s. York has a private left luggage place near the station - go out of the main entrance and turn right.

The left luggage place is a bit of a walk and not sign-posted. It is in the Europcar office.
See Google streetview here: https://goo.gl/maps/PiH9MbSp6TbXLHMk9
You have to go down the stairs where the man with the red case is. He obviously is either on his way to or from it. And two other people with cases walking in that direction.
The station roof is visible behind the black car.
And yes, that is York city walls on the right of the photo.

After you've dropped your luggage at the storage office, there will still be a good walk to the center of the historic district. Give serious thought to how much you'll actually be able to see in a few hours. Realistically, I think you'll lose about an hour between dropping off the luggage, getting into central York, retrieving the luggage (a partially-uphill walk) and getting back to the station in time to catch the train to Edinburgh. I enjoyed York a lot, but much of it is overwhelmingly touristy, and this plan would put you there during the busiest part of the day. Do you have plenty of time in Edinburgh, or will you lose (for example) 3 important hours in Edinburgh in order to scrounge 2 inadequate hours in York?

As said above, it can be done. It is up to all of you to decide if you should.

The very cheapest tickets are called "Advance". Advance is the name, and advance is when you must get them, weeks or months in advance. By far the cheapest, they are valid only on the exact train named on the ticket, and on some Advance tickets they are only valid in the exact seat on the exact train. That means no changes and no refunds (in some circumstances an Advance can be changed in the station before boarding the train and before the train is scheduled to depart, for a large fee - so in practical terms it is effectively non changeable, and never on the train or after it has departed). So using one Advance ticket what you want is impossible. If you decide to use back-to-back Advance tickets the challenge will be to predict exactly how much time you and your friends will want in York. If you guess wrong you will either run short of time or be killing time waiting for the planned train. Advance tickets to be good to your wallet must be purchased weeks ahead.

There are other tickets which are usually more expensive but which do allow what is called "break of journey", for example some "Off-Peak Singles" or "Off-Peak Returns", or "Super Off Peak Returns". Some of those allow a break of journey on the Return leg only, some allow it on both the Out and the Return legs. Each company has different restriction on different tickets so you will need to ask and check carefully. These are walk-up fares so don't go up (except a couple of times year when they rise with inflation), but equally they never go down. They are valid at certain times of the day, usually not at rush hour, and the restrictions are more extensive on Super Off Peak.

The most expensive tickets are "Anytime" and they do what it says on the tin. You can always break the journey on those. But you're paying through the nose.

I think I just have a personal mission to go back to York and go to Betty’s Tea Room because I totally forgot about it when I was there. I happened to walk by it after I just ate my last dinner of my stay there and was so bummed that going there completely slipped my mind! I did stop by and pick up dessert but it wasn’t the same.

I will have to discuss this with my friends but knowing them, they’ll want to go for it!

Chris F. Thanks for the google map link. We’re spending 3 nights in Edinburgh so I think we can spare a few hours in York but who knows? Our itinerary is still in the planning stages.

Nigel, thanks for your train expertise! I really was hoping you would chime in. We will go ahead and buy the cheap advanced tickets once we have our plans together.