There are usually a number of entertainers on the Royal Mile which young kids enjoy. The Museum of Childhood and West Princes Street Gardens are good options along with the National Museum of Scotland.
I think having fairly easy access to your accomodation would make your trip more enjoyable for both you and your child. A quick trip out to Rosslyn Chapel would be my limit for a day trip.

The main reason I recommended Linlithgow is it is a very short train ride, there are wonderful ruins to clamber over, and explore, very easy walking around the loch, feeding the swans and ducks, renting a row boat, a couple of playgrounds . . . all within a 20 minute train ride.

I missed janisj's suggestion of Linlithgow. Yes, a very good choice and the short train ride might be a new experience for a 3 year old.

I like Janis's idea. But I am wondering if they are willing to rent a car, what about the south end of Loch Lomond?

I went on a boat tour there from Balloch. I thought it was beautiful.

Thank you everyone for your input. We have finally decided to stick to Edinburgh and do a day trip to Loch Lomond and Glasgow instead. We will be renting a car as we have a toddler.

You mean Glasgow and Loch Lomond the same day? If so reconsider - Glasgow with car not fun and though I found it interesting in architecture and a pleasant city why spend your valuable time there - just to a more relaxing day at Loch Lomond and may hit Stirling and its awesome castle instead?

Why Glasgow?

NOT a god idea -- really. OK -- Why Glasgow? Glasgow is a great city but you seem to want country/scenery. Glasgow ain't that.

First of all a car will be a HUGE hassle in Glasgow. And Edinburgh to Glasgow, up the west side of Loch Lomond and back through the Trossachs to Edinburgh would be a 5 to 6 hour drive without a single stop. So a minimum 10 hour day with the hassle of getting around Glasgow and visiting anything there.You simply have no time for a driving tour anywhere. Honest. Renting a car for one day will be another hassle. You need to get out to EDI, collect the car, drive where ever you decide (NOT Glasgow), return to EDI, then travel back in to Edinburgh. All with a 3 year old.

Either do JUST London (with an easy day trip or two); JUST London and Edinburgh; or JUST Edinburgh and a few days touring Scotland.

I have been to every corner of where you are considering, some more times than one can count, and what you want to do just won't work.

I would not do London and Scotland - especially with a 3 year old. That is a lot of travel with a small child who probably still takes naps, will be bored to tears to be on a long train or car trip.

Personally I would do London and a day trip to Hampton Court, Oxford, or the Cotswolds,

If by train maybe overnight train would be best for kid that age - would probably sleep like a baby! But if kid can take 8-9 hour plane ride guess could take 5 hour daytime train ride but a lot of travel.

I would NOT take a 3yo on a London > Edinburgh sleeper train. If she doesn't sleep neither would anyone else in the car.

If you do go to Edinburgh, what about a day trip to Stirling Castle? Or to the Falkirk Wheel?

A day trip to York is great too. York has the walls you can walk about and fun snickleways although probably not easy to navigate with a stroller.

London has so much to do and I highly agree that a day trip to Hampton Court Palace is wonderful. The kitcken, the gardens, the maze, etc. etc.

There is the London Zoo too--small but easy to spend a couple of hours and that is in the city.

Don't forget the Tower of London. We spent about 5 hours there when we visited. There is so much to do there and you can take a boat ride over to Greenwich from the Tower Pier. You can visit the Tower Bridge observation deck.

Thank you for the feedback. Quick question on London tourist attractions- would you recommend getting a london pass?

Absolutely NOT. It is a huge rip off for most visitors. They overwhelm you with the number of 'covered' attractions. But it is almost humanly impossible for someone to race around fast enough to visit enough of them in the limited time to pay off.

A much MUCH better deal would be the 2for1 scheme https://www.daysoutguide.co.uk/2for1-london

It actually costs nothing extra -- but you need to buy paper travel cards at a train station (instead of an OysterCard at an Underground station) to travel around on the buses and Tube

Ditto on the 2for1 vouchers...a great deal.

I usually say that the one attraction you should not miss IMHO is the Tower...though not so sure with a 3-yr-old. But you should research it and consider it.

I have to say that traveling with a 3-yr-old is a whole different thing. We did Paris with our 3-yr-old grandson (&his parents), and you really have to allow a lot more time to do everything than you would if you were by yourselves.

ssander