El Cid's #20 also describes the potential downside to "influence", which otherwise enjoys popularity as being accurate and already used commonly in this context. Influence contrasts with control: it sounds weaker because it implies you _might_ affect behavior, or you might not. "Power" suggests a guarantee of the result you want (if you've got enough).
I'm stunned that "non-military power" and "civilian power" have come up but no one's mentioned "peaceful power" or a variation. Maybe other people think "peace" sounds too weak also, but I think it's got rhetorical weight. No one calls himself a warmonger; no one says they're against peace. Opponents could imply "peaceful" means unwilling to use military force, but I don't see that gaining traction.
I think Matt's second notion -- the "appeal of the brand" -- can be summed up as "Respect". Respect, like "political capital", can be thought of as an element of peaceful power (look! it's catching on!). However, it's a cleric in the Peace Power party: respect doesn't make other nations do what you want directly, but it makes the diplomatic resources, development aid, etc. more effective. (Curse you, pseudonymous in nc at #24, for planting the RPG analogy seed!)