This is easily google-able and a matter of record. Quotes from Alinsky's book are readily available and can be verified. The answer is very clearly:
- no, he did not dedicate his book to Lucifer (or Satan), and
- no, he was not her mentor.
Two good looks at this are from Snopes and the Washington Post.
On the claim that he dedicated his book to Lucifer:
Both sources point out that he wrote about Lucifer being the "first radical" in the book's introductory pages, but NOT it's dedication page. From Snopes:
...while it's true that one of three epigraphs on an introductory page
(not a dedication page) of Rules for Radicals characterizes Lucifer as
the "first radical known to man who rebelled against the
establishment," the book is neither dedicated to Lucifer, nor need it
be read as an endorsement of devil worship or Satanism.
Here is the text from Alinsky, which is clearly not anything related to devil-worship, but is establishing a mythical context for radicalism (a la Paradise Lost):
Lest we forget at least an over-the-shoulder acknowledgment to the
very first radical: from all our legends, mythology, and history (and
who is to know where mythology leaves off and history begins — or
which is which), the first radical known to man who rebelled against
the establishment and did it so effectively that he at least won his
own kingdom — Lucifer.
You can read it for yourself.
On being her mentor:
Clinton wrote her undergraduate thesis on Alinsky. According to this New York Times article, she agreed with some of his main criticisms of (then) contemporary progressivism but advocated more for change-from-within than radical agitation and direct action.
Clinton has praised some of his work, but, as the Post concludes:
But there's little evidence that Clinton was particularly close to the
man. And indeed, her decision to write a thesis involving Alinsky
wasn't her idea, her thesis adviser recently told The Washington Post.
Alinsky had at one point offered her a job, which she declined. Ultimately, there is very little tying them together, other than an analytical research paper she wrote about his work when she was in her early 20s and a few subsequent exchanges.