David,
I love the idea of giving our staff opportunities to "practice Influence"!  I brainstormed (Ok, maybe it was a quick sprinkle, not a storm!), and came up with my ideas for "practicing influence"... Let me know what others you have!

1) Have them lead a meeting
2) Partner them with a new employee to show them the ropes
3) Before critical meetings with clients, assign them parts of the project or task to explain to the client during the meeting
4) Work with them to lead a staff training time or staff meeting

Any other ideas?