My theory for "why" Kevin did it.

Now, I don't think there is any one specific motive that lead Kevin to to the thing on "Thursday", but i think there is one explanation that has been neglected that I think is plausible (I'm basing this more on the book than the movie):

1. The book heavily implies, through various bizarrely erotic undertones, Kevin masturbating in front of his mom, etc. that he suffers from a sort of "Oedipus complex", in which a boy is infatuated with his mother, not necessarily in an overtly sexual way but at least in an emotional way. His antics can plausibly be attributed in part to this.

2. He ends up fulfilling the Oedipus fantasy by murdering his father.

3. Unlike his father, Eva was left alive by Kevin, and Kevin refers to her as being his "audience".

4. Kevin is strongly implied to have an obsessive crush on/be in love with student Laura, one of the victims of the massacre . Eva speculates that "for all I know he was trying to impress her".

5. Acts of extreme violence such as what Kevin did are often speculated to be the result of teenage boy's desire to prove their own masculinity in an over-the-top way. Kevin clearly thinks of the mass murder game as a rivalry, as evidenced by his contempt of the CHS shooters.

So, I think, whether Kevin knows it consciously or not, what he did at his high school was in order to impress Eva as young men sometimes try and impress females with acts of violence. Killing Franklin and Cilia was his foremost objective and would have been enough for him, the thing at his high school was really just for show.