Please allow me to make a small observation regarding Lily’s point about the lack of recognition amongst students of “well-known” professors in a given HE institution (HEI).
Beyond the more direct link between the output metric of more publications in higher prestigious journals which helps an HEI on some academic ranking tables, there is another reason well-known professors may help their students directly. To publish in a renowned magazine, a researcher needs to be fully conversant with the state of the art in their field. Such a researcher will embody the most current information available. (Not withstanding the imperfect relationship between being a good researcher and a good teacher), this knowledge is physically present at the HEI. Teaching at that HEI will be positively affected by that knowledge, either directly in the researcher’s classes or via their teaching assistants (TA), many of whom will be their graduate students.
Even if an undergraduate has never heard of a particular ‘name’, that ‘name’ will have an immense and relatively-speaking immediate effect on the quality of information and knowledge available on campus. Furthermore, not only will that affect the quality of the courses, it will impact on the reputation of the HEI leading to higher quality entrants (via more competition) into the undergraduate courses, and perhaps to more engagement amongst the students, TAs and less ‘named’ research staff as they have a personal, local benchmark to aspire towards or compete with.
Of course, as Lily rightly points out, if that name is used only nominally (pun intended), the good deriving from it will be limited.