"My hampton Bay three speed ceiling fan works fine on high and medium, but on low, it barely rotates and I can't really hear the motor running. It is run by pullchain. I opened the housing and the black box, which may be a capacitor, shows signs of melting. Would this cause a problem with only the low speed and if so, is it easy to replace?"

Rather easy, simply take the unit down and throw it in the bin, go out and buy a new unit.

There's a reason that the unit is showing signs of something melting... it's faulty and for the price of a new unit you will save yourself time and money.

Capacitors are replaceable and much cheaper than an entire fan. You need moderate knowledge to do the swap

There's only one way to find out - replace the capacitor and see what happens. There still may be an underlying problem that caused the capacitor to burn out in the first place, I would assume that replacing the capacitor may "melt" it again or cause some other problem within the unit. I do realise that replacing the unit is more expensive than replacing the capacitor, but peace of mind is what I'd go for.

I hear you but capacitors frequently fail just because they want to, nothing else is bad to make them fail, they just do it all on their own

In that case, go for it - it shouldn't be too hard to replace the capacitor if you've got the right equipment (which I assume you do). Again, my only concern would be the fact the it's melted/burnt, "usually" when they fail, they don't burn out, but not un-common to do so. Good luck with it ;-)

Is it a remote fan?

General ceiling fan troubleshooting and help FAQ- Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-general-troubleshooting-and-FAQ.php)