Seems like there's been quite a few different picks on this question. (Not too many people have voted on this one, but I bet if we got together all the old fans and they voted, the variance would be even greater.) I love series where you have so many interesting characters that there's a lot to choose from besides the main ones.
I guess I already said Artemis and Holly (maybe leaning toward Holly, though as I said, I go back and forth), but after that...
Maybe Minerva third. I know fans can have a negative association with her (because in fanfiction, having a new OC girl who's also a genius feels cliche and unrealistic), but I did feel like her introduction was done well, and I didn't have a problem with it. And I do like the idea of a female genius--I remember reading Death Note (which features quite a few characters with genius-level intelligence), and feeling kind of let down there weren't any. (Although, maybe part of my problem was that I felt like Death Note had a bit of sexist undertone running throughout the series... x3 The types of roles all the key female characters played...) She has a pretty interesting arc throughout The Lost Colony, and I liked how it mirrored Artemis's through the earlier part of the series. Although, maybe the reason I like her is the feeling of her potential as a character. (Which of course we know was never really realized.)
After that...possibly Butler. I'm often a sucker for characters who are intensely loyal (part of the reason Misa-Misa was my favorite character in Death Note--psychopath that she was, I remember being impressed by that gratitude and loyalty to Light, even in confinement and under intense interrogation...but, anyway) and Butler and Artemis's relationship is one of the things I loved about the series. (I remember really being drawn into that first part of TEC. When I was first reading it, I really didn't know if Butler would actually survive, and that made it a pretty heartwrenching set of scenes.) I also love how Butler's skills and military sense of efficiency compliment Artemis's particular skillset so well, and really, as impressive as Artemis's intelligence make him, it's really generally Butler who makes Artemis's brilliant plans possible.
Strangely, Opal also became a character I really liked over time. (When I initially read the series, I guess I didn't particularly care about her one way or another, and I did sometimes feel like a lot of fans, 'Not her again.' xD) But as I read the series more times, and really began to see all the details that made a real foil for Artemis ('foil' is a literature term I heard in high school; it refers to a character whose similarities to another character help accentuate the differences). Not only are they both geniuses and criminal masterminds, but they've grown up in similar social spheres, with the privilege of wealth. (As Opal says, they might have been allies if they had met under different circumstances.) But, Artemis clearly cares more about the people close to him, while Opal doesn't really have anyone. I loved the contrast that was set up in TAI, where we learn of how Opal basically ruined her father, in contrast to Artemis, whose primary goal throughout the book is to save his father.
“After all, absolutely no one can help but suspect a criminal, liar, and manipulator of committing crimes, lying, and manipulating. And of course, no one is more aware of that simple fact than Artemis Fowl.”
Opal sets into motion her most diabolical scheme yet, to frame Artemis and turn his closest friends against him. Only this time she has a new calculating partner who knows Artemis better than he knows himself. [An Artemis Fowl fanfiction, set after The Atlantic Complex.]
...Shameless self-advertising, guys! C;
(And if you're really bored: AF fanart. ;J)