Well I noticed that the more obscure Toa teams have been excluded, such as the Hagah (my personal favorite) and the Mangai, etc. The exclusion of these bring up the question how do you define best? If split into story and sets like @Scorpion_Strike did, then typically the newer the sets mean the newer parts, armor/weapons, and redesigns, allowing for greater function and detail, meaning the most recent wave is the best.
Story-wise is much more difficult to answer and it goes back to the obscurity of some Toa teams, like Lesovikk's, Jovan's, Hagah and Mangai. Does less story coverage mean that these teams are worse?
The Hagah once successfully raided the Makuta Fortress of Destral, beating the Makuta on their home turf, an achievement that nobody has mentioned directly, and even as Rahaga, they overcame their physical transformation and helped the universe against the Visorak and other beasties. Toa Jovan's team also saved Mata Nui from dying, sacrificing one of their members like Matoro @Hutere, yet they aren't even considered. The Toa Mangai saved Metru Nui(the brain of the robot) from Dark Hunter takeover and the Kanohi Dragon, but aren't seriously considered @Mar .
And with regard to the Toa Metru it can be said that they had one major failure, which was to "save the heart of Metru Nui," or the Matoran, which they had to resort to plan B(get them out later when they were in the spheres), allowing Makuta to cause the Great Cataclysm and shutting down the entire Great Spirit Robot. Despite failing to prevent the rise of Makuta and failing to prevent the Great Cataclysm, which would later lead to Mata Nui dying, the Toa Metru are still considered story-wise as one of the better teams because of their development in the story, that is, they became Turaga.
The point being, best story-wise can't be determined because, of the vague meaning of "best." Does best mean more story coverage, more importance to the story, more development in the story, or more achievement in the story? Because a team is less famous than the Toa Mata, Metru and Inika, or is deceased does it mean that it can't be considered as best?
A more specific answerable question would be which is the Most Powerful Toa Team?
Edited for Double Posting - Waj