How should a Christian put a decision in God's hands if there is no logical answer? Lets say a Christian is in a position where he must make a hard decision (assume it has to either be a yes or no by a deadline) and each one has negatives consequences (more negatives than positives). How should a Christian pick one option over another? 

Both options can and will affect the future of himself and others, but it is impossible to tell which one will impact the least or which one is a better decision, only time will tell.

Based on Christian teaching, if Jesus specifically was talking to that person, how would he advise, or what are some things he would tell the person to keep in mind?

I am sorry it is not specific but I can't put details to it. So if it helps, the main part of the question is how should a Christian put a decision in God's hands if he really doesn't see a logical answer but has to pick one? (I am also assuming the person was put into making that decision by God?... since he would rather not be in such a situation...)
Proverbs 3:5-6 says 


  Trust in the Lord with all your heart and lean not on your own understanding. In all your ways acknowledge him and he will make your paths straight. 


Elsewhere in Proverbs (16:9 / 20:24), it says


  A man plans his path, but the Lord directs his steps. 


The consistent pattern I keep seeing is that if we are faithful to provide the motion, God is faithful to provide direction. Indeed, the case of Jonah, even when Jonah was not faithful, God used Jonah's movement but ultimately steered. (I've been known to say, God will gladly kick the sides to steer, but not the butt to move!)

The point that I'm making (and it would be consistent with Rom 8:28 as well) is that God will ultimately use your faithfulness to accomplish what he desires, regardless of whether or not you made the "right" choice.