To eye someone has apart from the meaning of look at the sense of doing this with a certain amount of suspicion. To me I eyed her closely suggests almost that you are watching someone either with the in kind of surveillance way or because you are trying to see any reaction she may display. Let’s be specific. Imagine you have reason to suspect that she is not telling the truth. You ask her a question and while she is giving her answer, you eye her closely to see if there is any trace or sign that she is not telling the truth.