For the most part it's not exclusive so much to Sailor Moon, although this represents an extreme case of editing. When Sailor Moon was first brought over to the United States, the people who had the American distribution rights decided that it would be unsuitable for kids to have a foreign cartoon. That's how anime was viewed, as a kids' cartoon, although in Japan the definition of anime is much different. 'Anime' is the shortened version of the English word 'animation' and oddly enough, 'anime' is a foreign word in both the USA and Japan. XD
Anyway, what happened was that in the process of dubbing, all references or hints at Sailor Moon's Japanese origin were scrubbed, and questionable lines deleted. For example, all of the Japanese signs and onomotopoeia were removed; the original, unedited version clearly takes place in the Juuban district of Tokyo. ('Juuban' just means 'tenth', Tenth District. They like to be vague in anime when it comes to cities and locations.)
Along these lines, then, the names of all the characters were changed most likely because they were too Japanese and unsuited for American children. Those that could be adapted without too much trouble were; others were just invented offhand. Note that Chibi-usa Tsukino's real name is Usagi Tsukino and that she's called Chibi-usa to differentiate; you lose this subtlety in the edited version. The situations and characters change, too. Some episodes are stricken entirely because of questionable practices that would probably raise protest in America; in the second episode of the original, Usagi's brother Shingo kicks a dog. In the first episode, Usagi as Sailor Moon is attacked with a broken bottle; the bottle is edited out in the American television version. Additionally (here you can have some photographic proof) a little bit of fancy dancing was needed with some lines:
In one episode the girls are arguing over who ought to play Snow White in a production. In the dubbed version, Lita suggests that she ought to play because she has the most talent. Compare that to what Makoto says in the unedited version and I think you'll see what I'm getting at,
here.
To answer the original question, this practice of renaming characters has declined drastically since Sailor Moon. Later series that were widely aired such as Gundam Wing, and even those that have been edited extensively like Naruto usually at least preserve the names of the characers. Only the worst 4Kids hatchet jobs would probably change names now.