When Shakti took the form of Parvati, she was a Deva: the adopted daughter of a mountain and a being which is best understood by the word "nymph" (an "aspara," the spirit of a cloud or water). She was one of three sisters: Her adopted sisters devoted also themselves to Shiva, and to demonstrate their devotion ate only a single leaf for their food every day. This represented the very minimum required for life (for a creature like they were: the cross-breed of a mountain and a nymph). But Sati knew Her true nature (as a Deva - and as Shakti), and thus she required no food at all - She only required the love of Shiva. She gave up eating entirely, to gain the attention of Shiva and demonstrate Her true nature to him. Her starvation gained the attention of her parents and sisters, too; astonished and overawed, though they had known she was a Deva, they did not really understand it - now they understood her true nature, and even saw she was Shakti, Love itself.
She was first called by them Aparna, one who eats nothing. Then they called her Uma! The name implies a combination of "Oh, mother!" "Oh, beautiful!" (Shakti is a "mother" of all beings, and is "beautiful"), this is an exclamation similar to the English "Oh, don't!" (Parvati's austerity seemed unnecessarily cruel, but in fact was required to gain the attention of Shiva, and preserve Parvati's life). This was an exclamation of beauty, splendor and tranquility - in so many more ways than these, the name represents the moment when her family understood her true nature.
"Uma" is also, however, a masculine term for a wharf or landing place - when it is remembered that Shakti is partially female and partially male, partially Shiva, the concept that She was presenting a landing place for Shiva is also understood better in the context that She was presenting herself to Shiva, not so much as an offering (how could She offer herself, when She was partially Shiva?) but in the sense of permission and availability. As a wharf or landing place may be beautiful enough to entice a traveler, She appeared so beautiful.
"Uma" is the state of being her sisters were attempting to achieve, but which only Parvati might.