QUESTION: Is it true that a nakshatra is the area of the heavens transversed by the Moon’s movement within a day?
That’s right. That is the basis for dividing the night sky (which is full of stars) into 27 equal sections. (because there is a little more than 27 days in a lunar month, there is an extra Nakshatra too, Abhijīt – which does not really fit neatly in with the others.).
The rāśi (zodiac signs) are similar: the area of the heavens traversed by the Sun in the course of a month is the basis for dividing the daytime sky (which is full of sunlight – raś) into 12 equal sections.
QUESTION: Then what is the relevance of the individual stars associated with the nakshatra?
Nakshatra’s divide the sky into 27 equal vertical sections, along the horizontal line of the Moon’s orbit. Within each section is a constellation of significant spiritual importance. Each of those constellations has one or two primary stars. These constellations define the nature of the 27 divisions they fall into. They do this because they are associated with Vedic divinities. The Vedic divinity associated with a star is the primary determinant of the nature of the entire nakshatra area that star-group belongs to.
QUESTION: How do particular stars come to be associated with particular deities?
Ṛg Veda has given us the association. I am not exactly sure why. I can make guesses or give ideas about some, but I do not have a conclusive, consistent understanding for each star.
QUESTION: Why don’t the Nakshatra follow a standard of nomenclature? they appear to be named after either the Deity, the associated star or a symbol such as a hand in the case of Hasta.
The Vedic texts call them directly by the names of their deities.
The interpretive Jyotiṣa texts follow the nomenclature standard common to all astrological symbols, naming them in a way to encapsulate a description of their nature.