Get the smallest soldering iron you can. Put a dab of new solder to get a nice shiny look on the solder. Just DON'T DO WHAT I DID. Make sure you have a desoldering tool, in case you accidentally join to solder joints, if this happens DO NOT POWER IT UP. YOU WILL KILL THE ANALOG BOARD. Get a Desoldering tool and undo the mistake. I destroyed my Classic 2 this way.

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I would offer quite different advice. To get this particular connector to solder properly requires a fairly robust iron. Small, low-wattage types will struggle to get the pins hot enough to solder well. A cold joint is a poor joint, so you can't skimp. Indeed, the high failure rate of this connector's joints is attributable to the high thermal mass, and the consequent inability of the original factory equipment to produce a good joint in the first place.

Also use solder fairly liberally (not so much for the solder as for the flux). The surfeit of flux will assure that oxidized residue will not get in the way of making a good joint. After removing the iron, verify that the joint stays smooth and shiny as it cools. Try not to disturb the joint mechanically as it cools, or you will vex the magical metallurgical gnomes that control the interface.

Then, to avoid the catastrophe that Mars478 alludes to, simply inspect your work carefully before re-applying power. One would think that this step hardly needs explicit mention, but data suggests otherwise.