My first suggestion is to start team battle tourneys. These could be held periodically in a round robin or elimination fashion. Prizes could be distributed based on placement in the tourney.
I also think that 1 or more consolation prizes that is equal or near equal to the winning prize could go out randomly to the "losers" of the tourney.
Another game mechanic that could make things interesting would be HQ mods. Just like our toons have gear slots, why not have gear/mod slots for the HQ? Mods could have different qualities like trophies w/o the penalty. This would allow teams to customize their HQ, giving them advantages they don't already have. Mods should also be stackable. These qualities could be highly varied. Examples follow below.
% increase to one or more stats for team members.
They could mimic trophies.
They could mimic boosters.
They could mimic sidekick skills.
Other suggestions welcome.
