Choosing a thesis advisor is probably one of the most important decisions you make as a graduate student. Here are a few pointers that might help you out.
When: The choice of thesis advisor is required to occur by the last day of finals of the spring semester of your first year. Some faculty may give conditional acceptance depending on things such as passing a Qualifying Exam or passing certain courses. Such conditional acceptance by a thesis advisor satisfies the requirement of choosing a thesis advisor. If other issues arise, please speak with the GPD as soon as possible. The GPD can give an extension for finding a thesis advisor, depending on the nature of the circumstances.
How to Pick: The first thing to consider is research area. Choose an area that is interesting to you and one in which you do well. In terms of interest, in the long run, you usually come to love the field you study so it probably does not matter too much which area you pick—mainly avoid areas you do not enjoy. It is probably most important to pick a research area in which you have a lot of success.
Another thing to consider is personality, temperament, and working style. Ask other students about the professor you are considering as a possible thesis advisor. Listen to their Pizza Seminar talk, take a class from that professor, do a reading course, ask for some papers to read, or sit down and visit with them. You will spend a lot of time with your advisor so make sure that sounds like something you would like to do.
Procedure: There is no formal procedure or form. Simply ask. A bit more explicitly: If you think you want to work with Professor X, ask Professor X if there is a time that you can meet together to discuss the possibility of becoming their graduate student. If that sounds too intimidating, a nice way way of getting your foot in the door is to ask Professor X for a paper to read.
You should bear in mind that some professors already have graduate students and therefore may be unable to take on additional students at this time. This is no slight to you, but simply a matter of making sure that they have enough time to devote to their current students. It is also possible that various other extenuating circumstances might prevent a professor from accepting students at a particular time. Again, this is no slight to you.
If you have a thesis advisor and wish to change to another one, that's fine! You may switch any time you wish.
However, when you pick a thesis advisor, you and your advisor must assent that you have read the Mentor Agreement. This is required. Students initiate the electronic agreement HERE, and research advisors are sent an email to complete their portion.
Thesis advisors will submit a yearly evaluation of their thesis student's progress.