The freezer is where I always keep my ginger (root) --and also my hunk of Parmesano Reggiano.
It's a great solution to always having them around! I keep each one in a ziptop baggie.
There's no nutritional loss at all, really ever (though may eventually get freezer burn and not taste as good after a long while).
The texture can be a little different but ginger is usually grated or use in slices so there's no detectable difference (and the same for Parmesan, though I admit I don't use it for just eating a hunk--mostly used for cooking and on top of all kinds of things).
For the ginger, you will usually want to peel the brown skin (if you care about looks) before grating or slicing. I usually do that right before using a sharp knife (straight down onto a cutting board) or sometimes scraping with the edge of a spoon. I usually just do an inch or so at a time, which isn't difficult even though it's fairly frozen. (And I usually freeze the whole piece since it will also be easier to use the rest of it as a "handle" when grating or peeling.)
There are actually all kinds of "sometimes-things" that can be kept in the freezer. For example I freeze lime and lemon halves. The limes will eventually get a bit brown looking and even a bit frosty, but the juice tastes just fine.
I do the same thing to ginger when I have a lot. I do nor believe the nutritional benefit is lost immediately upon freezing, though if you have year old frozen ginger, then obviously in that time it would have lost it's flavour and nutritional potency. I think you're best with freezing it up to 3 months.
You can freeze it, one would assume that freezing it does effect texture, flavor and nutritional value, since it effects everything that way. I've seen/used freezer products that have fresh herbs (like ginger and garlic) in frozen form, they're okay, though you end up using a bit more than you would fresh. I'd put it in little piles on a a plate or in a spare ice cube tray, freeze it and then put the frozen piles in a ziplock. I don't use it for ginger myself, but I've done lemon/orange zest this way and it works well.