What would cause the capacitor (black box) to melt in the fan control box? I have a friend who has a Hampton fan and called the company about her switch not working, and they sent her a new switch at no charge. (I found the burnt capacitor, and after that she insisted on getting the new switch to hope to solve the problem.)
When I return to install it, I wonder if this will solve it. I'm thinking that it's a burnt motor and told her so, but she insisted on trying the new switch. So, to summarize, what could cause the capacitor to melt or burn on one side of it? Am I right in thinking that it's a motor problem? Thanks!

If the fan was used with an improper wall control or dimmer, that would cause it.

Sometimes they just melt for no reason. Happens pretty often.

Ceiling fan capactors - troubleshooting and resources - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-capacitors.php)