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Probably from the old Sanskrit root R which means to go or to rise (it is actually Indo-european and some of its cognates also have meanings like to row or to plough), one gets Urmi' (related to words like English well) meaning sudden appearance, speed or wave. It is often used figuratively for the six waves of pain (hunger, thirst, decay, death, grief, illusion) or for the six waves of existence (cold, heat, greed, illusion, hunger, thirst), or more mundanely for wave like folds of cloth, for other straight lines, or for almost any strong emotion compared to a wave. In technical contexts, it can also stand for the number six.

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UrmilA, as you point out, is an old name, and is still in use. It means one possessing Urmi, and it is usually parsed as ocean, especially associated with the connotation of ocean of emotions.