How important is Master thesis when applying for a Ph.D.?

Sometimes choosing a famous name backfires. If it's a hot shot with tons of students, he will distinguish among his students, when writing his letters, by giving strong letters to the strongest among his flock, and mediocre letters to the less spectacular. Such a mediocre letter from a big name can be quite damaging to an application.

Even aside from that -- choose the person who fits best:

• you are interested in their work

• you admire the person ethically

• they bring out the best in you

• their management style suits you

When I began working on my Master's Thesis in an engineering field (in the United States) I asked my advisor a similar question, namely: "Should I perform a problem report or a thesis?". My advisor, who is a well established and active professor, told me it depended on if I ever thought I would do a PhD.

He said that a Master's Thesis and the associated research were essentially training to learn how to research and document effectively and efficiently. Therefore, a well researched and written thesis is very important for admission into a Doctoral program - because research is what a Doctoral program requires!

Of course, as stated in another answer, having a well-known advisor could be a double-edged sword in its own right.