I would agree with malmostoso's idea of finding a flight to Europe you like and then adding Athens as a 'domestic' flight separately. Airlines like Easy Jet have very low fares within Europe you could take advantage of.

One thing you don't say about this trip is how much time you will have and what you want to do with your time other than visit Athens. I have difficulty accepting that just any city will do as an add-on. It has to be a place that is of interest to you all. I would also suggest that adding a city may not be a good idea at all depending on how much time you have and what your children's interests are.

You also don't say how old the kids are. The younger they are, the more important the length of a flight becomes. Shorter being better obviously. Keeping a 5 year old occupied en route for 20 hours is harder than with a 15 year old. So that would make the shortest possible flight distance to Europe appealing and I would certainly not want any connecting flights such as those shown in malmostoso's example with a near 7 hour layover in Heathrow giving a total travel time of 21 hours from SFO to ATH. I would consider a non-stop essential. Price is not the only factor to consider, no matter how important it is, there is comfort to consider as well as you say.