Sorry, haven't been at the computer much the past few days. I'm sure I'm forgetting some, but here are a few examples that might usefully placed into an underground parts of plants category in the database:
Rhizome
Taproot
Corm
Bulb
Tuber
There are at least two types of tubers, stem tubers and root tubers, but I'm not sure it is worth making the distinction? Interestingly, potatoes (stem tubers) and sweet potatoes (root tubers) are two easy examples of the difference between the two types of tubers.
Stolons are a bit of a conundrum as well. They typically grow along or above the surface of the ground, so wouldn't seem to fit in this category. But, stolons are basically pretty similar to rhizomes. Both are stems that grow horizontally and produce new roots and shoots at nodes along their length.