First off, the question of origin for such a colloquial phrase is a hard one to answer. Oral language is often only documented after it has been in use for some time, and anything before the first documented use is going to involve a lot of guesswork. This answer will give (a) predecessors that used similar "whatever verbs you/your thing" phrasing and (b) documented uses of "whatever floats your boat."
Whatever verbs you(r) noun
Fairly late in my research, I turned to the Corpus of Historical American English, since Google and academic search engines weren't giving me much to work with. After a couple of tries, I hit on a collocation search for "whatever _v" (_v is "verb") and "you*" (you, your, young - the asterisk is a wildcard). I found a lot of results.
Most results were questions: "whatever are you looking at?" and "whatever gave you that idea?" were two of many examples. There were also uses that functioned as objects of verbs: "He will pretend he is curing you of whatever ails you."
However, there were a few results that functioned quite like whatever floats your boat - uses that defer to someone else's preference. The first is quoted in full as it appears in the database, since the play may give more context for the usage. Then two more entries follow.
John Howard Payne, Richelieu, 1826.
Janet (aside to madam Dorival) Don't say a word, madam.
Mad. D. Madam Dorival (apart) Just Heaven! What shall I do!
Rich. Duke of Richlieu (artfully) Ah, I see! My company would be
Dor. Dorival (to his wife) What ails you, my love? Is this mere
absence of mind? Sure you'll welcome our friend Lamotte once more to
Mad. D. Madam Dorival (forced to speak, but with her eyes cast down)
You know -- whatever gives you pleasure -- (turns abruptly away, and
says aside) -- My heart revolts from what I say!
Rich. Duke of Richlieu (with pretended frankness) Enough. I thank you
for your invitation, and will not fail to be at supper.
Jan. Janet (aside) If he does, may it choke him!
Dor. Dorival That's kind, -- that's friendly now; and, to make the
party more pleasant, I'll have a friend to meet you
Anna Maynard, The Award of Justice Or, Told in the Rockies A Pen Picture of the West, 1897
" All right, Mr. Blaisdell, that will be perfectly satisfactory, whatever suits you young fellows, suits me. "
Alice Skelsey, The Working Mother's Guide, 1970
Now you have all that time to laze around, catch up on projects, slip in a mini-trip or two without the children? whatever suits your purpose, just so it is a satisfying change of pace.
So the pattern of saying "whatever suits your purpose" or "whatever suits you" was already in English by the time the 1973 blues album Whatever Turns You On came out, let alone the other forms of "whatever turns you on" that show up in the 1970s.
Whatever floats your boat
Despite all my effort, the earliest source I can find for the usage is
in the Oxford English Dictionary under "float, v.":
transitive. colloquial (originally U.S.). to float a person's boat: to
interest or excite a person; to appeal to or suit a person. Esp. in
whatever floats a person's boat.
1981 Sunday Herald (Chicago) 16
Aug. v. 7/4 Venus enters your house of travel on the 18th to stay
until Sept. 12, so make getaway any way you can. Fly, drive, row or
read. Whatever floats your boat.
1989 T. Parker Place called Bird
xxi. 259 Whatever floats your boat, you've been doing it all your
1991 Blitz Sept. 103 Over the following four pages..is a
composite of the men and women who float our boats and why.
Midwest Living Apr. 51/1 (advt.) There are plenty of opportunities
for fishing, swimming, hiking, biking or whatever floats your boat.
2002 More! 3 Apr. 97/4 Small breasts don't float my boat.
In the OED entries, by the early 1990s the phrase has become more flexible: "who floats our boats and why" becomes possible. Green's Dictionary of Slang notes the extended usage "float one's boat in" as early as 1984, where it appeared in Connie Eble's chapbook Campus Slang:
1984 Eble Campus Sl. Sept. 3: float my boat – stimulate, excite: That don't float my boat.
Connecting the Dots
Given these instances of usage, there are three parts to this guess of an answer:
"Whatever verbs you(r) noun" appeared in a few forms as a statement that defers to someone else's preference through the 19th and 20th centuries.
"Whatever floats your boat" appeared as a specific instance of this linguistic meme. At the very least, the resemblance is there.
The float / boat rhyme generalized in use far beyond its "whatever" origins. The similar meaning and preservation of the rhyme suggest that much.