Yeah, as it's been said, Mothra, Rodan, and King Ghidorah are set to appear in the Legendary Godzilla movies, and in likelihood, it would probably be something like Ghidrah the Three Headed Monster, that idea just sounds like it would make the most money and since there is no real word on when the movie is coming out, it would be better if it was something like Ghidrah, since if the movie comes out in 2018, then people would need a reason to see a sequel to a movie that came out four years prior and it would be a post, Batman v Superman/Avengers 2 world, movie audiences would need something more than just Godzilla, sad to say, however, how can Legendary write the monsters they already have, especially Mothra and King Ghidorah? Well lets get started and see.
This would be the easiest monster to write so let's get this guy out of the way, Rodan I see being an animal from Godzilla's time period, back when the world was full of radiation, and Rodan would be not as much as a prey animal for Godzilla like the Mutos were, but more like a competing predator, much like how wolves would compete with bears for food or something like that, so they wouldn't exactly seek each other out, but if they ran into one another, Godzilla and Rodan would battle over dominance/territory, ability wise, no big change from his Toho counterpart other than making him Fire Rodan like in Godzilla vs Mecha-Godzilla 2, and give him his own atomic breath, it would make sense giving that Godzilla is a naturally occurring animal in this movie's universe, so having another creature with a similar ability wouldn't be too much to stretch in the imagination.
Mothra is probably the hardest to do since she's a mystical creature and the Legendary Godzilla movies are going for something of a more realistic take on giant monsters(as realistic as you can get with this subject mind you) but I think I've got an idea for her, and the idea is this, Mothra's ancestors existed during Godzilla's time and instead of going into hibernation like the rest of the creatures, Mothra's kind evolved over the eons into feeding on solar energy from the sun, and were almost wiped out when the dinosaurs died out when the asteroid hit, leaving only one Mothra survive, and Mothra in this universe would be asexual, able to reproduce by herself and lay one egg, so over time, only one Mothra existed, and when humans evolved, they worshiped her as a god and there's stories about her fighting different monsters throughout different cultures with the last one dying out around the time of the Romans, leaving an egg that gets buried on an island where that Mothra lived, the natives worshiping her, around the time when the male Muto woke up and went to Japan at the beginning of the first Godzilla, he uncovered the egg and it gathered solar energy until it hatches almost twenty years ago, this Mothra once an adult has a modified version of what an electric Eel has and can shoot electricity from her antennae, the Shobijin, would be really different, being twins that were born on the same day as when Mothra's egg gets uncovered, and are raised as priestesses, they also have telepathy, which they learn to use to communicate with Mothra, and making her actively help humanity
This is also a hard one, on one hand you can go with the alien route and have Ghidorah come to earth by some means and throw the whole ecosystem out of whack, making Rodan, Godzilla, and Mothra all attack him, or you can make him a genetic experiment, Gareth Edwards said that he'd want to later down the road have it where humanity would try to create their own monster, making Godzilla the new atomic weapon, with this I can kind of see, now hear me out with this, I can see Ghidorah be a genetic experiment where scientist try to splice DNA from Godzilla and Rodan, having wings similar to Rodan, but a bulk like Godzilla, the golden scales, long necks and three heads/two tails would be a mutation that scientist couldn't work around, with this, Ghidorah would be kind of unstable, and hyper aggressive, attacking everything in site and actively destroying everything from cities to forests, and again, forcing Godzilla, Rodan, and Mothra to team up to fight him, I'd also have there be a splash of Mothra DNA as well and have Ghidorah feed on solar energy as well as nuclear radiation, and his gravity beams would be a combination of Godzilla/Rodan's atomic breath and Mothra's electric attack, in that it would be a combination of something like fire and lightning
Listening to: Godzilla by Blue Oyster Cult
Reading: Godzilla: Half Centry War
Watching: Linkara's History of the Power Rangers