Lugia could put up a good fight against Mewtwo. But in the end Mewtwo's psychic abilities are far greater than any other known pokemon. Plus with such a large move pool (Icebeam, thunder, blizzard, shadow ball) his victory is almost certain. However... A lugia with the ability Multiscale (which halves attack if his health is full) could focus his power with calm mind. Raising his ressitance to Mewtwo's massive special attack. If Lugia holds out long enough. And then recovers his health with roost. Lugia might be able to lauch an offensive (X4) that is more over powering then Mewtwo. This tactic probably won't work though as Mewtwo has both calm mind for focusing his energy up to 4X and recover.
Mewtwo would usually win against a Lugia. Unless it is a Lugia that can keep up its defence long enough. And hopw mewtwo stays on the offensive.
In the games, it is clearly stated that:
1.) Lugia's Special Defense is equal to Mewtwo's Special Attack.
2.) Mewtwo's Defense Stats are equal to Lugia's Attack Stats.
3.) Both have equal HP.
But Mewtwo has a little more Speed and access to slightly to moderately stronger super effective attacks like Thunderbolt/Thunder and Ice Beam/Blizzard thanks to Lugia's additional Flying Typing.
I'll give this to Mewtwo for those reasons.