To me it can mean either that the decision could have gone one way or the other or that it was very carefully considered "weighing up" all possibilities and outcomes.
2 Answers
If a decision was finely balanced, it was difficult to decide (generally between two things). Two or more arguments were convincing, and thus the correct decision wasn't immediately clear.
Because such a decision requires parsing the pros and cons of different arguments, it's probably more carefully considered than most. Whether or not this makes it a fair decision, I think, is debatable.
Not to me. A finely balanced decision is one in which the arguments for and against it were of almost equal merit. A decision in which all aspects were carefully considered is a fair decision.