I see what you mean. I tried both the Finnair and the American websites and came up empty. There just are not a lot of links to Minsk.
One very useful thing I learned from a post on this forum is that if you go to the Wikipedia page for an airport, you'll find a table listing the airlines flying out and their destinations. This makes it possible, when necessary, to figure out a way to pair an inexpensive trans-Atlantic flight with a European flight to the desired destination. This must often be done on separate tickets, leaving you twisting in the wind if something goes wrong on the trans-Atlantic flight. I wouldn't be comfortable with a same-day onward flight in your situation. Even next-day could turn out disastrously in the event of an aviation strike or a volcano eruption in Iceland, so this is something to be considered only if it's really, really financially advantageous.
For Minsk I see that Austrian Airlines flies in from Vienna; LOT, from Warsaw; Lufthansa, from Frankfurt; and Turkish Air, from Istanbul. I was hoping to find single-ticket itineraries through those cities.
Warsaw has one advantage in that it has bus and train transportation to Minsk (7-1/2 to 10 hours), which could be used in a pinch. I took a look at Lot.com. Unfortunately, there's no provision for doing a multi-city routing, at least not that I could find. Very strange. I looked at one-way fares Atlanta to Warsaw out of curiosity and found that the (very high) one-way fare is $400 lower on June 21 than on June 22. That might be useful to keep in mind as you do further research. Unfortunately, as the schedule currently stands, flying LOT to Warsaw would mean 10 hours at O'Hare, which would be a miserable way to start the trip.
Using Frankfurt as the European gateway seems as if it might be the least expensive option. On the Lufthansa web site I was able to book the round-trip you want, albeit at a cost of almost $1900. The outbound leg was a non-stop ATL-FRA on Lufthansa followed by a non-stop on Lufthansa Cityline (a commuter airline, I think). On the return I was hoping for a Rome-Atlanta non-stop, but such was not to be. All the offerings involved links on commuter airlines.
I next tried Austrian Airlines, but I got the "cannot be booked online" message. You could try calling, if only to see what the routing looks like.
Turkish Airlines. Wow. I found a price of $1010. Pretty unbelievable. But there are a couple of issues: There's a 22-hour layover in Istanbul outbound (after a connection in Toronto) and a 14-hour layover on the way back. And, although I would have no safety concerns about this routing, I think we can assume some of the children's parents would freak about the time spent in Turkey.
The Belarusian airline, Belavia, has flights into Minsk from a number of potentially useful European cities. Amsterdam, London-Gatwick and Paris-CDG are the ones l think likeliest to allow you to find a bargain-rate trans-Atlantic flight.
I know nothing about Belavia and would tread very, very cautiously there. Some of the airlines in territories formerly part of the USSR have very poor safety records. I'd do heavy-duty checking before flying with Belavia. Full disclosure: I don't ever fly US commuter airlines, so I'm hyper-cautious about this sort of thing.
How flexible are you on your dates? Could you leave one day earlier, fly into Warsaw (if LOT would allow multi-city ticketing) or Kiev, and use a pre-arranged van transfer to Minsk? That would avoid the need for a risky, separately-ticketed flight connection into Minsk. The most direct route from Kiev to Minsk is about 360 mi., but ViaMichelin estimates 9:40 without any stops, which I guess tells us a lot about the road... Please note that I know nothing about road safety in that part of Europe.
Vilnius, Lithuania is substantially closer to Minsk and has several trains a day, but it is 2 stops from ATL.
