Now the important bit, the new capacitor. It's got to be the right one for the job, not all capacitors are the same! Instead of throwing specs at you (that are printed on the capacitor body) I will suggest a specific part number.
The stock capacitor is usually a Nichicon brand A9531. It has since been discontinued. Good riddance, it clearly wasn't that great. However, I'm told Nichicon is a good manufacturer and so my replacement was another Nichicon and ordered it through Digi-Key, which is a great seller. Their site is a little tricky so just put in the part numbers and go that way.
Digi-Key part number: 493-11653-1-ND
Manufacturer part number: UPV0J121MFD1TD
This is one of Nichicon's long-life models, made for way more cycles than the standard line capacitors, and the size is almost identical to the original capacitor (a hair taller but fits fine). Overall it should be a much longer life unit.
My cost at the time was about 54 cents per capacitor, so why not splurge and get two? Just in case you lose one or something.
The capacitors come packaged like this, just pull one off and trim the leads down short so it won't be sitting up high.