How much direction do you want?
Some advisors give you a well-defined thesis-sized problem, explain an approach, and tell you to get to work on it.Other advisors are hands-off: they may give you no help in choosing a topic at all, but can be extremely useful to bound ideas off of once you find an idea.
So do you work better with a lot or a little structure?
How much contact do you want?
Some advisors meet weekly, possibly in groups, for progress reports. Others meet on a demand driven basis. Some might want to meet once a semester.
How much pressure do you want?
How much emotional support do you want?
Who will act as a filter for ideas?
Some advisors toss out lots of ideas and expect you to figure out which ones are good ideas. Other advisors will try not to overload you with ideas.
Is the advisor willing to supervise a thesis on a topic outside his/her main research area?