Chibiusa's full name as presented by the manga is Princess Usagi Small Lady Serenity. Now unless I am utterly mistaken, in the anime they never refer to the "Serenity" bit, although she does get Usagi, Small Lady, and Princess. From the name Usagi we get Chibiusa courtesy of Mamo-chan, and that's where we hit out first fanon pothole. So, how exactly do you write Chibiusa? It can be written Chibi Usa, Chibi-usa, and even Chibi-Usa, if you want to stretch it. You can make a case for writing the name any of these four ways. Chibi Usa might be correct because when Mamoru names her he intends her to be "Little Usagi" thus justifying the space because they're meant as two separate words. Just the way you'd write something like "Lil' Usa" in English, and indeed, Chibi Usa is a pretty popular way of writing it on American fansites. Sites that use this way of writing the name often also use "Sailor Chibi Moon" for similar reasons.
Chibi-usa and Chibi-Usa get dragged in because a hyphen is an easy way to compound two words that don't necessarily go together (although this isn't used very commonly in names outside the joining of two family names). Other fans are fond of writing it this way and "Sailor Chibi-moon" often finds its way into use on these sites as well.
I personally use Chibiusa and Sailor Chibimoon consistently because I think of Chibiusa as one name, her first name, even if it is a nickname. You can see it rather like the name Maryanne, which is separate from the name Mary Anne because the first is simply a first name and the second is first and middle. The Japanese are also very fond of slurring words like that together when they write things in English on merchandise. Thus you see a lot of "Sailorneptune" and "Sailormoon." I do like this way of writing it out, but "Sailorchibimoon" looks a bit much to me, so I keep with "Sailor Moon" and "Sailor Neptune" to be consistent with my "Sailor Chibimoon."
In the end, you really shouldn't worry about it. No matter if you call her Chibiusa, Chibi Usa, Chibi-Usa, Chibi-usa, or that Pink Brat, people will probably know who you're talking about. I just thought I'd clarify my opinions on the matter.
Now that given names are aside, Chibiusa's last name (which she takes on in the past) is definitely "Tsukino." No, it's not "Chiba," not even in the future (where you can argue that her family name is the name of the dynasty -- "Serenity"). In Japan it is custom that the more prestigious of the family names become the name of the family. Thus husbands sometimes take the names of their wife's family, as do the children. This way the primacy of the family name will continue to flourish. It is unclear whether Mamoru takes the Tsukino name in the future before they ascend the throne, but Chibiusa certainly takes it when she's in the past because, well, the female heir of the Tsukino family is now Queen of the Solar System.
Now onto further ticky details. Small Lady is a title and not a name (not even Mamo-chan would be that cruel). Presumably it is her title as princess and heir to the throne of Crystal Tokyo, likely given (because it originally comes from the manga) because she did not grow at all for nearly nine hundred years. Being nine-hundred years old and still trapped in the body of an immature nine-year-old is certainly one way to be classified as a Small Lady. Small Lady is also what the folks in Crystal Tokyo call her commonly, just as they refer to the queen as Neo Queen Serenity and not "Usagi" or even "Queen Usagi." Perhaps in very close family circles they might refer to each other by their first names, but this doesn't seem to be true in either the anime or the manga. When she's in the future it's "Small Lady" all the time, whether she wants it or not. Rather like the "Prince of Wales" only more prevalent because people also call Charles "Prince Charles" as well.
I don't really need to go into Serenity much, other than the fact that in Chibiusa's name I see it as a dynasty identification more than anything. In the Silver Millenium everyone called Princess Serenity "Serenity" or "Princess Serenity," and Chibiusa doesn't get that. It's always "Small Lady" for her, although at a stretch it could also be "Small Lady Serenity" I am sure. For Chibiusa, it seems mostly there just so everyone will know that she's Serenity's heir. It's the same reason she takes the name "Tsukino" in the past.
As far as the dub (and the American translation of the manga) goes, I honestly wish they'd gone more along the lines of the French manga and left her some name like "Mini-Bunny" which is about as close to "Chibiusa" as you can get in English if you're going to go changing things in the first place. Of course, the domestic release of the dubbed anime came far before the domestic release of the translated manga so to try and keep some consistency (and not confuse the fans, whom presumably they thought to be puerile idiots who could not take a name shift) the domestic release of the manga kept with the dubbed name.
"Rini" was chosen presumably as a diminutive of "Serena," Usagi's dub name. Why they chose to spell it "Rini" when it's supposed to be a diminutive of a name with no i's in it at all, I have no idea. Perhaps they thought it was cute and more trendy. Things that end in 'i' usually are. Fans still persist with the spellings "Reenie" and "Reeny" though, and I can't say I blame them. That at least makes a little sense.
When Chibimoon finally appeared in the dubbed anime there was some debate over what she would actually be called. She had been called "Chibimoon" when she appeared in the translated manga earler because no precedent had been set. Fans seemed fine with it, so many people hoped that Rini might actually get to henshin into Sailor Chibimoon. Other people thought that if the name had to be changed to accommodate people who didn't like any Japanese at all in their dubs, then Sailor Crescent Moon might perhaps be an acceptable change. Although not close to the specific meaning of "Chibimoon" Crescent Moon at least keeps the similar sound and similar meaning. A crescent moon is only a quarter of a moon, isn't it? Well, some fans liked it but the board who made the decisions didn't, so we ended up with "Sailor Mini Moon." Praise god we didn't end up with "Sailor Rini Moon" which is rumored that they debated for a while and then eventually abandoned. That would've been so much worse, trust me.
"Sailor Minimoon" had already been in use in the French manga, so you can't exactly give that board of executives a prize for coming up with it. In the end it fits the bill pretty well. Mini is a relatively close approximation to Chibi. My major beef with "Minimoon" both at the time and still today is not so much the name itself but how it fit into the new direction they took with translating and rewriting Rini's lines and how the actress chose to portray her. In the S and SuperS dub, Rini has very little of the spunk and brattiness that she had in previous seasons. She is almost purely sugar sweet and wonderful, sugar goodness love love heart heart. It's enough to kill you with the Barney-esque message. When she first arrives, Sailor Minimoon introduces herself as the soldier of "Love and Togetherness" as opposed to "Love and Justice" as in the original. The name "Minimoon" just seems to feed into that idea of Rini as the perfect sugar princess. Now I like cute and I like pink, but making Chibiusa into just pink and cute and nothing else is doing her a huge disservice. There's so much more to her: courage, passion, and fire, and none of that is present in the dubbed "Minimoon."
But still, it's a lot better than "Rini Moon" isn't it?
The last thing I can get out of the way is my interpretation of the naming of the Star forms of the Sailor Senshi which they attain just after the defeat of Neherenia (in the manga, these forms were never attained in the anime). Some people refer to these forms as the "Eternal" forms, after Sailor Moon's Eternal form, which is similar in design. When the senshi are called out in their Star forms they are referred to as "Sailor Star Jupiter" and "Sailor Star Mercury" and so on. Chibimoon is referred to as "Sailor Star Chibimoon" much in the same way that "Sailor Star Fighter" is named. It's my opinion that Star as now become a part of the senshi's names now that they've attained this form. Of course, they can also still be referred to by the shortened planet name (like Pluto or Chibimoon) but the actual name of the form itself seems very kin to the Super in "Super Sailor Moon." It's the actual name of the form that is consistently used, not just a sparkly add in to make the senshi feel special. Use the naming of the Starlights for an example.