Depending on the airline, fees for taking your bike on the plane can be anything from pretty reasonable to insane. It depends on the airline. But looking around earlier this year, it seemed a little less insane/variable than a few years ago. Last year flying to Italy with my bike, I lucked out on a cheap Business Class fare on Condor. It wasn't that much more than a Coach fare once I took into account my bike flying for free--that was a pretty sweet experience!

From my research, a quick pros/cons--
Hard case--sturdier but heavier, expensive to buy, may be hard to keep box+bike under weight limits
Soft case--lighter but a little scary to trust your bike to something with soft sides, good ones are also expensive
Cardboard box--cheap and light! Least complicated disassembly/packing experience. Bulky and unwieldy as heck. Can be difficult to fit into smaller European cars.

I was going to say that if someone brings your bike on a flight with them, it better be someone who really loves you--but then I saw that your mom might be the one flying over Flying with a bike is definitely a pain--even the best luggage cases are bulky, heavy, annoying things to have to maneuver through an airport and transportation. But if your mom is willing to deal with the hassle, the cardboard box might be a good option. It's cheap, and actually a pretty sturdy way to pack a bike for a flight, especially with generous use of bubble wrap. And you can kind of push/slide it along the floor in the airport (stairs suck though).

FWIW, my husband and I both travel with the Pika Packworks EEP bag. I like it a lot (though it would be nice if it had wheels--on the other hand, it doesn't have wheels to be broken off, which seems to happen a lot with wheeled cases.) We've also traveled with his bike in a cardboard box.

Some good info in this article and the links in it: http://cyclingtips.com/2014/06/flyin...ggage-handler/