In addition to what's been noted above, you should bear in mind United's policies regarding one-way and roundtrip awards. (Other airlines have significantly different policies.) UA offers one-way awards for half the mileage of a roundtrip: 30k in each direction for the trip you have in mind. The advantage to one-way awards in your case is that you can book the outbound trip now, instead of risking that saver awards will become unavailable by the time the return can be booked. A disadvantage is that one-way awards do not permit a stopover, as RT awards do. However, even though you mention a stopover in your inquiry, the trip you envision does not involve a stopover.

If you have difficulty finding 3 saver awards on the same flights, you might consider travelling separately and using different partner airlines on the same day. Obviously, this is less desirable and may even be impossible depending on the age of your son. But United has several partners, such as Austrian, Lufthansa and Swiss as well as Turkish, that serve Athens and/or Istanbul, and it may be easier to find 1 or 2 award seats on a given airline than it is to find 3.