I think the Articuno should be named "Adriana", for being the primary and most complex personality of the trio. The Zapdos should be called "Plasma", the state of matter embodied by electricity. And, finally, the Moltres should be christened "Corona", after the Sun's halo that is responsible for solar flares.
If I'm not too late, I'd like to try my hand at this too.
Articuno: Halla, a Finnish word a weather phenomenon where the temperature drops below zero during the growth season. It's usually very beautiful when the whole world seems frozen over and all the plants and trees turn silver. I think Articuno's presence might have a similar effect if it were real lol.
Frosti, a Björk song, pretty self-explanatory but still very beautiful and fitting.
Zapdos: Volta, a Björk album this time, and again pretty obvious, but ever so cool and fitting.
Moltres: Helia, IIRC a daughter of the Greek sun god Helios.
Aska, Umm not so sure about this one's origins. I've just always loved the name Aska and ever since playing Tales of Symphonia I've thought Moltres should be named Aska. There's this Spirit of Light named Aska who is the spitting image of Moltres. Light can also be taken to mean the sun which obviously means infernal fire and is very fitting for Moltres.
The reasoning behind my nicknames for Pokemon and naming things general is pretty simple. I want something really cool, beautiful and different, but something that still actually sounds and feels like a name, is easy to pronounce in most languages and fits its bearer's personality and characteristics.
My favourite combo of these is Halla, Volta and Aska which also compliment each other remarkably well. Oh my, I might actually have to NN my next competitive birds Halla, Volta and Aska.
Articuno - Aurora Bird (Because of the Aurora Boreau on the Arctic regions)
Zapdos - Shazam(The powerfull Captain Marvel from DC comics)
Moltres - Krakatoa Bird(Because of the Volcanic Island) or Phoenix because of the Mythical Bird)
All names related to their major color and all are nicknames of fictional characters known for their speed. Moltres' nickname is based on Char Aznable from Gundam, known for his red Zaku that moves three times faster than a normal Zaku. Zapdos' nickname is based on the Fourth Hokage from Naruto who moved so fast that he was nothing more than a flash of yellow. Articuno's nickname is based off Sonic the Hedgehog who is known as the Blue Blur.
If its still going...
Articuno- Chione, which is from the greek for snow, after a water nymph that created it, and still has the number one in it strangely enough.
Zapdos- Capt Hammer, after both the god thor's weapon that represents the thunderbolt, the "good guy" from dr. horrible's blog.
Moltres-Xiuhtecuhtli - aztec god of fire just sounds really cool,