I am not sure what causes this issue, but I have been able to recreate a workaround to this problem. Sometimes the accept button simply does not register that I have clicked it. The only way that I have been able to work around this problem is clicking the "Ask Question" button on the top, then clicking the back button on my browser. When click the accept button again, behold, the check mark becomes green with the answer accepted.
Running Windows XP Pro
Browser is IE 8
Other buttons work like "This is a favorite question."
Refreshing the page does not work to this problem.