For each potential advisor X, do the following:
- Make sure you don't have a problem with X's research interests.
- Make sure you can get along with X.
- Talk to X's current students about X's pros and cons. Ask clever questions since many students refrain from discussing the downside of their advisors with a stranger. Don't forget to ask if X is too (dis)engaged with his students. This step also helps you find out more about your potential academic siblings.
- Find out what project would fund you. Is it a long-term or a short-term project. If it's a long-term project, make sure you like it.
- Find out if X has a tenure, in tenure track, or neither. Non-tenure (track) faculty are more inclined to leave their academic institution and find a better position somewhere else, which is -usually- a bad thing for their students. Tenure-track faculty are more inclined to work harder, and tend to be more demanding (which can be good or bad, depending on who you are).
- Collect some statistics about X's research group. How many students does X advise? A very large number (e.g. fifteen) usually means X won't have time to interact with you, and a very small number (e.g. zero/one) usually means X has problem with funding (or a new hire). Find out what's the average lifetime of X's students in the PhD program; the variance is usually small.
- Find out if X collaborates with other academics (inside or outside the department/school). More collaborators usually mean a well-connected advisor, which can be very useful for your marketability.
- Find out where does X's graduates go.
- Meet with X. Ask smart questions. Be ready to talk about your previous work.