How do you preserve fresh ginger?
If your goal is longer term preservation, freezing is one of the easiest methods and what I tend to do on occasions when I've bought a lot of ginger  at once (for whatever reason).  While you'll lose a little flavor with freezing, when used in cooking, I find it still works well and is often difficult to distinguish from fresh.  And it's quite convenient: you can actually pull out a frozen piece of ginger, grate off what you need, and just put the rest back in the freezer.

As a side benefit, peeling and grating frozen ginger can actually be a bit easier than doing the same to fresh ginger.  If you need chunks of ginger, though, you'll need to let it defrost a bit first (or, alternatively, cut before freezing if you have a standard recipe you want to use it for).