I think a good utilization of multiple dice could add the bit of extra depth in combat that I've been looking for without mucking up the Lego-ness of it all with too many stats.
Perhaps each hero's attack phase begins with a roll to determine class specific attack abilities (a ranger's range, the number of times the agile thief strikes, etc) and then there are subsequent rolls to determine the attack's effectiveness (in this case, the power of the ranger's strike or multiple rolls to see if each of the thief's strikes hit, and if so, how much damage they do).
Obviously this is a bit half-baked at the moment but I think I'll be working on a rule set based slot around multiple dice rolls to play test later today.