To keep it usable, you need to store it in a freezer safe, resealable container. Whenever you want to use it, you just need to take it out of the freezer and grate it straight from frozen.
Freezing Diced Ginger
The best way to freeze those little bits of minced ginger is to store the minced ginger in ice cube trays. As long as you make sure that the ice cube trays are clean, then you will end up with little cubes of ready-made diced fresh ginger to throw in any dish.
In the Dirt
Whether by submerging it in a pot of soil, or even a pot of sand, your ginger will then take root and slowly grow in the dirt, allowing for it to easily sit there without going off. As long as you use it within about 4-6 months to prevent it from starting to send out green shoots, you can easily store your ginger this way making it last longer.