The description is quite vague.
If I read the paper and I have questions about it, who should I ask?
That should be the first author and in most cases this will fit with the description of C.
The ideas of B are indeed important, but in most cases this is overlooked and underestimated. If B didn't publish his ideas or did anything about them, then A could claim he had the same ideas, and B only stated the evident (I have seen this many times). More often than not, the role of B is to appear in the acknowledgements of the paper, or not to appear at all, because the conversation between A and B may be considered either:
- as a personal favor
- an informal or non-profesional conversation
- some mentoring or help that is to be thanked but not part of the paper
- a combination of several of the above or other gray areas of not-real involvement in the paper and the "real work".
Basically B gave his advice as a gift to A, and now A has to decide whether this gift is to be returned or not. If it is returned then probably B will be the second author, if it is not returned, then B will not be an author. This basically depends on how A sees B, he may be an ally, an asset, a friend, a superior, a rival, an enemy, an annoyance...
I have discarded the option of B being the third author because I'm assuming A is more senior than B, if B is more senior than A then B should be the third and not the second author, this would also mean that B has more chances to appear as a coauthor. The reason is that the order from first to last author means the involvement in the work and the paper, but also on low-level work, while the last author is in the highest-level work, most abstract, theoretical, general, long-term-visioned, etc. Therefore the most senior is set to the last (even if someone else was less involved but involved enough to appear) to avoid suspicion on a less senior researcher (aka lesser being) giving high-level advice to a more senior researcher.
Most of the time the focus, the stress and the effort is not on how things are, but how do they look like. At least this is what I have seen up until now.
BTW: When I put examples I like to speak about Alex, Bernie, Cory, etc. I think this is easier to follow and I do also think that @order made a mistake between A and B. By your description, A seems to be more senior than B, the kind of senior researcher that manages human resources, that can pay C and that can tell C what to do. If that is the case, I doubt as well about the involvement of A in the paper and the work, while B and A can