In his article “Alice on the Stage,” Carroll wrote: “And the White Rabbit, what of him? Was he framed on the “Alice” lines, or meant as a contrast? As a contrast, distinctly. For her ‘youth,’ ‘audacity,’ ‘vigour,’ and ‘swift directness of purpose,’ read ‘elderly,’ ‘timid,’ ‘feeble,’ and ‘nervously shilly-shallying,’ and you will get something of what I meant him to be. I think the White Rabbit should wear spectacles. I’m sure his voice should quaver, and his knees quiver, and his whole air suggest a total inability to say ‘Boo’ to a goose!”
Wikipedia puts it this way: “Overall, the White Rabbit seems to shift back and forth between pompous behavior toward his underlings, such as his servants, and grovelling, obsequious behavior toward his superiors, such as the Duchess and King and Queen of Hearts.” For example, in Chapter 4 of AAiW the White Rabbit says to Pat, the gardener: “Do as I tell you, you coward!” and “‘Digging for apples, indeed!’ said the Rabbit angrily.” Whereas in Chapter 12 he says to the king: “‘UNimportant, your Majesty means, of course,’ he said in a very respectful tone, but frowning and making faces at him as he spoke.”
So, the White Rabbit is oldish (middle aged?), and changes between haughty and grudgingly fawning. He is near-sighted (hence the spectacles) and basically cannot see anything at even a short distance. He is indeed “myopic” both literally and figuratively.
As to the rabbit’s name, we have chosen to call him “Harold.” Most of the animal characters in AAiW are not named explicitly, being called what they are: Mouse, Lory, March Hare, etc. Bill the Lizard and Pat the gardner are exceptions to this “rule.” The White Rabbit is not given any other name by Lewis Carroll. The King refers to him as “Herald” when commanding him to call the first witness, so we have jumped on the name as a pun.