Hi. This looks like a great project and is definitely doable. There are three main parts to this project: Making the butterflies, Making the electronics to get the butterflies to flap, and then incorporating them all in to a dress.
For the first part, it looks like there are small lengths of Nitinol wire available on Amazon for fairly cheap. Start with that and see if you can make a butterfly. I’m thinking that you probably want butterfly wings that are made out of a regular material and hinged together. Then you would use a small piece of nitinol, probably bent in to a U-shape, to connect the wings in the ‘up’ position. Finally, you would connect a small battery to the nitinol wire, and it would straighten out, moving the butterfly wings to the ‘down’ position. It sounds pretty easy, but I’m betting this may be the hardest part of the project for you.
Next, you would design a circuit (basically an Arduino, a battery, and a bunch of wires) to activate the butterfly wings according to a pattern. This is the part that you are going to get the most help with on the Arduino forum. It should be noted that you don’t actually need the Arduino101 that was mentioned in your video. Basically any Arduino will do. With that in mind, I would recommend checking out the Arduino Nano because it is the smallest version of Arduino out there. This will be important when you need to fit everything in to the dress. At this stage, you will also probably want to find someone who can design and 3D print a custom box for you to put the controller and battery in.
The third part is actually fitting the butterflies, Arduino, battery, and a bunch of wires in to the dress in a way that they don’t show, that the person wearing the dress doesn’t get tangled up in the wires, and that distributes the weight of the battery and controller so it doesn’t pull down on one side of the dress. I’ve helped to make a couple of very custom dresses like this, and I can tell you that it is going to take a plan on how to run wires to all the butterflies while still keeping the fabric flowing and loose. Usually, the best bet is to work the controller and battery in to the lower hem of the dress, but you may also be able to fit it in on the back or possibly in to an accessory like an oversized bow on the front or the back. It really depends on the style of the dress that you select. If you are very good at sewing, you might try to make a custom dress yourself. If your sewing skills are only moderate then I would recommend choosing an off-the-shelf dress to modify. The benefit to choosing an off-the-shelf dress is that, if you totally mess everything up on your first try, you can just go buy another dress and try again.
So, to recap, your first step is to get some nitinol wire from amazon and try to make a single butterfly. Make the butterfly out of whatever you like, but connect the wings together so they can move (think of how the hinge on a door works). Then move the wings up and connect them with a U-shape piece of nitinol. Connect a small copper wire to each end of the nitinol, then touch the wires to a battery (like a 9V battery) and see if the nitinol straightens out and the wings go down. Disconnect the battery to make the wings go back up. This is going to be the tough part. If you can make a butterfly, then we can help you design the circuit and write the code to make a bunch of them work.
I hope this helps!