Mark Zuckerberg, CEO and founder of Facebook, made a very public $100 million donation to the Newark, N.J., school system. What would the rabbis say about making such a notable gift? Can it be seen as a way to entice other philanthropists to follow in his footsteps? Should donations be made publicly, or is this counter to Jewish values?

go for it!

Judaism has a long tradition of praising modesty in the Mitzvah of Tzedaka. The Talmud in particular states that one who gives charity anonymously is "greater that Moses" [Bava Batra 9b]. Following this and other precedents, Maimonides [Gifts to the Poor, 10.7] says that the second greatest kind of charity is when neither the donor nor recipient is aware of each other's identity.

However, as the Torah itself, and subsequent law, culture and archaeology all attest, Jews have always inscribed their names on the gifts they have given, both to the poor and to religious life, stretching back to the Biblical tabernacle, the Jerusalem Temple [see Mishna Yoma 3.10] and to synagogues ever since. 

The tension here is succinctly summed up by R. Moshe Isserles, the major early modern codifier, who wrote [Shulhan Arukh Yoreh Deah 249.13] that one deserves punishment for being a self-aggrandizing show-off in giving Tzedaka, but that one is nonetheless permitted to inscribe one's name on objects that one gives to synagogues. As R. Isserles implies, there is a big difference between claiming credit for a gift - which is permitted - and being a show-off about it, which is profane.

Moreover, I would argue that there is something actually good about publicizing one's gift if it will spur future giving by others. As the book of Daniel 12.3 states: "those who induce the many to give Tzedaka will shine like stars eternally." It seems to me that Zuckerberg's gift can have a tremendous positive impact not only on the recipients of his contribution, but to impress an excellent generous example on others

First of all, Judaism firmly supports that the responsibility of education rests upon the community (when parents are unable to fulfill it).  There is an excellent article explaining the Jewish attitude toward education under the heading of Talmud Torah in Wikipedia, of all places.  However, your question seems to focus on the funding.  Let me answer your questions in turn:

1) What would the rabbis say about making such a notable gift? Kol hakavod.  After all, the rabbis were the educators of their time.  Money to education supported not only their livelihood, but their life's work.

2) Can it be seen as a way to entice other philanthropists to follow in his footsteps?  That question goes beyond Jewish values, except to say that if it is an enticement, than by all means make the gift.

3) Should donations be made publicly, or is this counter to Jewish values? Ah, perhaps this is the heart of your question. What you are recognizing is the contradiction between the Rambam (Moses Maimonides') 8 levels of tzedakah and an analogous concept of Jewish donation we might call b'shem omro.  The rabbinic concept of b'shem omro was about attributing any teaching to the person who taught it originally, and it is a highly-held value not only by the rabbis, but by the intertextual world we live in on the web.  It is a bit of a stretch to apply it to donations, but there should be some name for the Jewish practice of putting a plaque on everything.  This Reform responsa on the (re)dedication of synagogue windows gets at a justification of the practice as people seem to give more when they can place a name (either theirs or a loved one's) where it can be seen. Since this serves the performance of a mitzvah, it is acceptable.  So, while it may not be the highest level of tzedakah to give in such a public manner, as regards the donor, the community may receive even more benefit by the example of the donor being made public.  As long as the mitzvah (in this case of education) is served, a little publicity is fine.