Looking back a few years at my comment...
Pure speculation, but I'd be inclined to think that the "original" lyric was something else -- perhaps "Susanna," as in one of the other versions offered by Hugill -- and that it was later cross contaminated by the Santianna chantey.

I have a new theory, though I still think the "Santy Annie" bit is cross-contamination or the imagination of late-period presenters (including Mr. Hugill and his Bible!).

I have been looking over the various original renditions on record. In this thread we have some texts. Davis and Tozer's (which is surely filled with a lot of nonsense), which FWIW has 'Rosie.' Whall has,

And a-way you santee,
My dear Annie.

And you know what? His is the only one that matches "santee" with "Annie." The others, if they have some form of "Santy," finish off with "honey" or something else.
(Hugill mixed Whall's version into his melting pot of ingredients from every book he'd read, so you'll also find a "Santy-Annie" in there.)
Nor is there any consensus that "Can't you dance the polka?" was the standard line. It is a variation, that was sung by some. Also common was the line of "they love us for our money" (or whatever)—which actually rhymes with "honey."

Alden, in his 1882 article, had:

As I was lumbering down the streets of bully London town,
I spied a Yankee clipper ship to New York she was bound.
And hurrah, you Santy, my dear honey;
Hurrah, you Santy, I love you for your money.

So, a few things:
1) "Annie" does not confirm "Santy" as "Santa Anna," because "Annie" is only attested the once.
2) "Honey" is the most consistently cited, and one can see how Whall's "Annie" and Bullen's "man" could be derived from that. FWIW, Alden thought the use of the word "honey" earmarked it as an African-American song—or, more accurately, a song with slang that he perceived as Black slang. He may not be correct, but i think these *perceptions* by contemporary observers are notable.
3) "Honey" helps to confirm the "money" one as the last line, at least in some common version. Though I can't say what happened, it seems likely that the "money" line came first, and then the "polka" one developed from that.

4) Some people would agree (as I think Q stated above) that the song has the feel of a minstrel song. (And I think this is essentially what Alden, who said the song was "unmistakably negro" was getting at; he made little effort to distinguish minstrel material from what we might call "authentic" Black singing.) But *which* minstrel song, if any, was it based in?

5) Even if "Santy" does not match "annie," and therefore was not "Santianna" split in two, what was "Santy"? Hugill's passage, quoted by Jerry above:

"The older Packet ship words were: Away you Santi, my dear honey... or Away you Santi, my dear Annie... Sometimes too one would hear 'Away you Johnnie, my dear honey' or 'my fair man' (Bullen), but in the main 'Santi' was sung. Now no one as ever given a real reason, or meaning, for this word; it just appears to be a meaningless name of some sort. I thought so too, until I came across a version giving 'Away you Santa, my dear Anna' and the explanation become clear – the mysterious 'Santi' or 'Santa' being nothing more than the two first syllables of our friend 'Santi-anna' or 'Santa-anna' or, as it was usually written, 'Santiana'!"

Now here is my suggestion for the minstrel song inspiration. Note that, per my view of chanties, the varying solo lyrics are not very important, because in chanties they would either be improvised or themes from elsewhere would be spliced in. Chorus and tune are what's important, with perhaps the solo lyrics providing a general "feel."

"Cynthia Sue" was a song sung by Christy's Minstrels. Mahar (_Behind the Burnt Cork Mask_, 1999) dates it to 1844.

Here's a link to one version of lyrics, as they appeared in _Christy's Nigga Songster_ (1850). It begins,

Long 'fore dis time, dis nigger dwell
In a place called Tuscanoe;
I loved a gal with tarry [tawny?] skin—
Her name was Cynthia Sue.

Oh, Cynthia, my darlin' honey,
Oh, Cynthia, I lub you more den money!


http://books.google.com/books?id=W2ZCAAAAYAAJ&pg=PA80&dq=%22Cynthia+Sue%22+chris

Here's sheet music of a LATER version of some sort. The lyrics have changed (in the chorus, notably), but you can see the tune of the chanty.

http://www.thehackley.org/viewer/?nam=Cynthia+Sue+%3A+as+sung+by+Christy's+Minst

And here's a rendition of this (1859 version?) by Timothy Twiss:
http://www.youtube.com/watch?v=a6PX-rzD1q8

So my theory....dun dun dun!.... is that "Santy" came from the two-syllable "Cynthia".