How do I make this Cyclops costume item?

It seems to be secured by the goggles. You’d need some way of clamping or otherwise fastening the goggles to the fabric, and the fabric would need a finished edge that could take the tension without tearing or pulling loose. I’m thinking you might need to turn the upper edge of the fabric in and run something through the channel (i.e., around the back of the head) that would attach to the inner side of the goggle earpieces. It might take some trial and error to figure out how to do that—especially with fabric that’s stretchy enough to move in.

You’re saying make the goggles part of the hood, so I can hide the headband for the goggles inside it, and also keep the tension on the spandex?

@Jeruba it turns out that’s exactly how he kept it on: Look here

I came to suggest basically the same thing as @Jeruba. I’d make a channel in the top of the ‘half hood’, and thread the strap of the goggles through it. You’d probably have to secure each end of the channel to the strap, maybe with glue or velcro, so the fabric doesn’t slide toward the middle.

And the goggles in turn are going to need some pretty good attaching to you. If you depend on your ears to hold that whole costume up against the tension of the fabric, they’re going to be sore pretty quick. It would be best if you included another feature—say, a narrow strap that goes over the top of your head, as close to your hair color as possible—to support the weight and stress of the material.

You might want to try this solution first in throwaway cotton or other cheap material just as a proof of concept.

In any case, be sure to reinforce both the top and the front edge of the fabric and especially the corners.

If this were my costume and I were planning to wear it for some length of time, such as at an evening event, one thing I’d do is incorporate some sort of built-in fallback in case it becomes too hard to sustain: for instance, making sure the neck portion can be detached from the goggles and rolled or folded down into a collar while leaving the goggles wearable and without the attachment apparatus showing.

Similarly, when I made a cat costume one time, I attached the tail with three snaps so it could be removed for sitting comfortably en route to the party.