I have a Hitachi Drive, but I'm 90% sure that that drive has nothing to do with it.
Basically, the thing ejects normally every other way, from the dashboard and manually ejecting. However, the button just does not respond.
Recently, the little piece that connects to the drive to press the button (It's a combination of green plastic, and silver plastic) fell off, so I popped it back on but alas, the button no longer works.
So upon inspection of the board inside the box, I found that the controller sync and the eject use the same type of button, but the eject was loose and seemed a little out of place, and the blue button seems loose. Feeble attempts at straightening it has failed. I believe that is the issue.
How did it just fall off? The outer shell keeps it in place unless you opened it up. If it was open you'll need to open it back up and line it all up and make sure the stick in is connecting with the button inside the metal cage before you put it all back together.
Ok, this is really tough to show but I'll try my best.
First, here is the picture of the front left, below where the disk drive would be:
See those 2 blue buttons? Those are it.
On the left, we have the one to eject the disk, it's broken
On the right, we have the one that syncs controllers, it works fine.
The button on the right has 2 positions, depressed, and out. When you depress it, it makes a click, basically, you can tell when it's pressed. Then it returns to being out.
The one on the left just sits in whatever position I leave it in, and jiggles a bit, like a really loose tooth or something, all directions.
Also, the thing was a bit out of position. The one on the right fits snugly into it's metal frame.
The bad one was back a little. You cant see it in the picture, but on the top left and right of the face are 2 little plastic dots, that fit into the front of the metal frame, but they werent in, it was tilted back. The frame was also off
Anyway, I used a pair of pliers to secure it in the right position, but the blue bit is still failing.
If you even just happen to know the name of this part, I'd love to know.
I'd just call it the eject button located on the motherboard. I'm thinking maybe someone in the buy sell forums on Xboxscene might be able to provide you with one.