We will have 3 days after the 7-day Istanbul tour ends. What suggestions would you make about side trips from Istanbul?
buy and open jaw ticket and fly into Istanbul and out of Budapest. You should be able to get a connecting flight for about $300 depending on season.
If you mean close to Istanbul, then I would consider 1) Princes Islands which are lovely and only a 2 or so hour ferry ride away - there are 5 (or so) in total and you can just hop from one to another - the big one is called Buyukada and 2) the city of Bursa - you can get there by ferry with another transfer to the center or on a day trip via bus. You can also easily spend 3 more days just in Istanbul alone and explore more nooks and crannies - the tour will only hit the most well-known spots, but there are plenty others in this city of 13 million. I spent over two weeks in Istanbul and managed to find lots of things to do every day, including exploring the Asia side, going to many museums (Istanbul Modern, Archeological Museum, Dolambahce Palace, smaller galleries) seeing the more modern part of the city north of the Galata bridge, taking a local ferry and stopping along the Bosphorus and the Golden Horn, Pierre Loti cable car, and visiting many mosques including Suleymaniye and Eyup, etc. And of course a Turkish bath is a must. I'm sure your tour guide will have excellent suggestions for you based on your interests.
If you want to stay in Turkey Cappadocia is great and I have heard good things about Safronbulu.