When the variation is not getting added to the shopping cart this is usually caused by a JavaScript issue.
In most cases the theme that is being used is not including the JavaScript files in the header.
One way to test this is to quickly switch to the default WordPress theme and test the variation to see if this problem is still happening.
Also please see the following article on the Importance of Using a Properly Coded WordPress Theme:
If the problem is still happening after the theme test please let us know and we will investigate it some more to suggest a fix.