In short, Chibi Usa means Small Rabbit. Literally. Chibi=small, Usa=short form of Usagi, meaning rabbit. But why was she given this name? Well, she was named after her mother, who's name is Usagi, so when she went into the past, she went by "Chibi," since she was the smaller of the two rabbits.
Her last name, Tsukino, means "of the moon." So when you put them both together (the family name always comes before the given name, in Japan), you have, "Rabbit of the moon." Pretty nifty, huh? Ok, well, we get the part about the moon......but why a rabbit? Why not girl? In Japanese legend, the "face" in the moon is that of a rabbit, not a man, or a girl. So Naoko named the girls what she did, and gave them a personality to match. Here is what an online baby names search had to say about the name Usagi:
"Your first name of Usagi has given you creative ability, imagination along practical lines, and the patience to pay attention to detail for a while. Although you are attracted to technical, mechanical, scientific fields, you lack the patience to follow through with this interest. This name gives a certain amount of practicality to your thinking; but there is also a tendency to scatter your efforts for, although you want system and order and stability in your life, you are too apt to be distracted from the job you are doing and to become involved in spontaneous interests. You are good-natured, have a good sense of humor, and are fairly easy-going, making friends with those who enjoy a good time. A weakness of this name is a tendency to make promises which you have difficulty in keeping. You tend to be lavish in your tastes and to spend your money freely. You have a heart of gold and are always ready to give generously. Since your feelings are strong and you tend to be enthusiastic and boisterous, you burn up your energies quickly and indulge in quick-energy foods, sweets, and starches. This type of diet affects the functions of the liver, causing either suffering with gall stones, jaundice, or skin troubles."
As you can see, this is very close to both Usagi, and Chibi Usa's personalities. Although it fits Usagi better, there is a little resemblance to Chibi Usa. It's interesting how it works out so well.
Again, Chibi Usa shares a name with her mother, "Serenity." This seems to be a tradition that is passed down through the generations. Queen Serenity named her daughter Serenity, who later grew up to be the queen of Earth and the Moon, Neo-Queen Serenity, who gave birth to a little girl she called "Tsukino Usagi, Small Lady Serenity." She never went by "Usagi" except in the past, but always her formal name, Small Lady Serenity, in the future, "Small Lady" for short. Small Lady, again refers to the fact that she is small, and young, and it is even mentioned by Sailorpluto in the R arc that she will one day be called "Beautiful Lady" when she grows up. Lady is a royal title, the feminine counterpart to the title "Lord."
What about "Serenity," though? We know that it is the tradition passed on from Moon queen to Moon queen, but why Serenity? Why not just Usagi? Well, there are different versions of this. It can actually be "Selenity," also. In Greek myth (which had a lot to do with the themes/symbolism in Sailormoon, see Symbolism), Selene was the goddess of the Moon. Then there is Mare Serenitatis, the place on the Moon that scientists believe was once a river. Serenitatis, meaning "Serenity." Because in Japanese, the "R" sound is exactly like the "L" sound, so Serenity would sound the same as "Selenity." Because of this, either way is acceptable.
We see that she has two names in one, but she only says the full thing in very formal occasions, such as when Black Lady says, "I'm your daughter, Small Lady Serenity," so why does she have the two names? Because she has Serenity, the name that seems to be given to all of the Moon princesses, and Small Lady to distinguish her from her mother, which would later change to "Beautiful Lady" (although we never see this), and probably later even be dropped.
Black Lady comes from "Small Lady," when she is transformed by Wiseman (see The Darkside of The Moon). She is no longer small, but she is evil, and since she is the same person, part of her name would be kept. Although it is not stated, Serenity was dropped, because it was a link to the Moon family, and because Black Lady is anything but Serene. ^_- Black, like Small, Lady, and Serenity, are all English words. The Japanese word for "black" is kuro, which is the name of Kurozuki Cooan, one of the Ayakashi sisters (see The Darkside of The Moon). Although traditionally in Japanese culture, white is the color of death and sadness, it is obvious by all the English names and titles, that Western culture is a large influence on the story. Lady, again, is a royal title, showing her connection to the Moon family, and the Dark Moon family, as queen of Nemesis. "Black Lady" could also refer to the fact that her heart is "black," lonely, dark, almost empty.