I think everyone has their own little definition of this term, and I don't disagree with ken's thoughts on the subject... To me, "thinking like a lawyer" really is about dispassionate analysis, emphasis both on "dispassionate" and "analysis."
Dispassionate: Don't reach a conclusion based on emotions, preconceived notions, opinions, desires, needs, or wants. Be outcome-neutral during the process. (If anything, favor the "bad" answer in law school and on the bar exam (purely as an exam/game strategy), as some professors use unpleasant facts to try to trick people into arriving at a morally acceptable result, and may give extra points for a correct-but-evil answer.)
Analysis: Favor the process over the result. You may never reach a firm conclusion, and that's ok. The process has to be thorough and methodical. If A, then B. But why not C or D? Identify all possible answers, eliminate when possible, and don't make any leaps of logic.
Various states publish a few answers to bar exam questions each year, to be illustrative as examples of good answers. Famously, many of these "good" bar exam answers have lots of wrong legal conclusions. Lots. But the analyses are good (and dispassionate), so the answers are good. (In practice, however, we do like to arrive at the correct answer as well as applying good analysis.)