See if you can get HEL or ARN both airports are good. And then you could use say Airbaltic to get them to BUD.
That would then involve 2 separate bookings, and if the flight from the USA is late the airlines are not responsible. It is an absolute must that your friend should book a through ticket on one reservation. I would also recommend you investigate routes where there are more than one flight from their transit airport to Budapest exists, then if they miss their intended flight there is another that the airline can put them on. Despite what people have said I would consider Heathrow as well as Amsterdam, Paris and Frankfurt, all of which will have multiple flights to Budapest, so will have the contingency needed.