There is a piece of hard rubber like material that you are pressing on that actuates the switch. It's not part of the switch. Has it simply worn down to where it is difficult to press. Or gotten soft from age and chemicals such as hand lotion and sunscreen?

If the switch below the button is still working, then you might be able to rig something. I've seen threads on Garmin Forums, BITD when they had the Edge 500 sub-forum, that showed users fashioning a make shift button to fit. How long it worked for them I don't know. Essentially you just will have to make something that will sit proud of the case so you have something to push on again.

If you do decide to get a new Garmin Edge, don't go for the models with buttons only. I made that mistake when I replaced my Edge 500 with a Edge 530. I found out that there are so many new features on the 530 that I'd like to have use of during a ride. However getting to them with button pushes was too intensive and required too much attention to get to them while riding. So I tended to ignore a lot of nifty little features.

Now that I have a Edge 1050 with touchscreen those features are a cinch to get to while on the bike and pedaling.