Had the idea vs. did the work: Who should be the lead author?
The lead author should be the person who carries out the bulk of the research work—including the planning, execution, and writing. 

If the student C is the one responsible for carrying out the experiments and writing the paper, then C should be considered the primary agent of the paper. 

Who should be the "final" author is a different matter. I would suspect that this really should be professor B, who had the original idea. The role of doctor A is only as research manager, so his role is nominally weaker than that of either B or C in this process. So, ultimately, barring other negotiations and arrangements to the contrary, it should be:


  Student C, Doctor A, Professor B.


If there is some other distribution of roles beyond what is laid out above, it would need to be negotiated between everybody involved.

Note: In fields where lexicographic order is not used, the final author position also carries weight. It usually is the head of the research group or team that originated the project.