Well, I've done a quick bit of research into a few areas to try and get the 'correct' answer in regards to the Pokémon 'HO-OH'. I've always used (and believed) it was 'Ho-oh' and not 'Ho-Oh', but that doesn't mean I was right, or wrong, for that matter.
My first idea was to look at how it's done in-game, but then I remembered that every Pokémon in-game is in all capital letters, so that got me nowhere. This is the case for every Pokémon, whether it's in an Item description, the PokéDex, mentioned by an in-game character, or something else. Do we really want to have to type every Pokémon's name out in capital letters for every article on Smogon? I don't think so.
Next, I took a look at some of the episode listings from the animé. This was also somewhat inconclusive, as whoever updated it on Wikipedia used 'Ho-Oh', while Serebii.net used 'Ho-oh'. HO-OH has appeared in only one episode title, 'For Ho-oh the bells toll'. Since I found one saying each of the current options, I can't be sure which is going to be right. Maybe if someone can find the official video or DVD release of this episode and take a look at the cover to see how it is on there. This is getting a bit obsessive for something so small.. but I'll keep typing anyway (lol).
The only other Pokémon with a '-' dash in its name is Porygon-Z. Every site I've read Porygon-Z's name on uses a capital 'Z', but we can't be 100% sure if that's right anyway, as it comes under the same situation as HO-OH really. But even then, that could just be because Porygon-Z only has one letter after the dash, not two or more.
In regards to typing, it's always '-type', for example: 'Fire-type' with the latter word, which is more than a single letter long, being in all lower case. However, when we look at moves, the only move to follow this suit is U-turn. Every other move in the game with a dash uses a capital letter, as follows: Double-Edge, Lock-On, Mud-Slap. Sand-Attack, Wake-Up Slap, Will-O-Wisp, and X-Scissor. Lastly, the only Item in-game with a dash is the 'Up-Grade' item, which also uses a capital letter for the secondary word. It's possible that 'U-turn' was an error, as when in-battle, and a Pokémon uses said attack, it shows 'U-Turn' instead.
So, despite me preferring Ho-oh, it seems that currently, there's more evidence to suggest that if Pokémon were typed in mainly lower case letters, it would be 'Ho-Oh' and not 'Ho-oh'. The only way to prove that it should be Ho-oh, in my opinion, would be to do what I suggested in my third paragraph.
So, there you have it, hopefully that shines a bit more light on the bird :)
Edit:
Actually, I found it. It clearly says 'Ho-Oh' on the cover, so I guess that's the end of that one.