What are the origins of the word "nice"?

Interesting question indeed!

It originally meant 'foolish, stupid':

1250–1300; Middle English: foolish, stupid

From Etymonline.com:

late 13c., "foolish, stupid, senseless

The following excerpt explains how the meaning of "nice" became changed so much:

"The sense development has been extraordinary, even for an adj." [Weekley] -- from "timid" (pre-1300); to "fussy, fastidious" (late 14c.); to "dainty, delicate" (c.1400); to "precise, careful" (1500s, preserved in such terms as a nice distinction and nice and early ); to "agreeable, delightful" (1769); to "kind, thoughtful" (1830). In 16c.-17c. it is often difficult to determine exactly what is meant when a writer uses this word.

Its meaning changed to become "agreeable" over a long period of time, and now, the reason for its connotation(wishy-washy sense) in modern times:

If any criticism is valid, it might be that the word is used too often and has become a cliché lacking the qualities of precision and intensity that are embodied in many of its synonyms.

It really depends on what you mean by "originally". We can trace the word back at least as far as a Proto-Indo-European form "skei" meaning 'cut', 'split' (i.e. about 5,000-6,000 years ago). This developed into various Latin words, including "scindere" (from which we get the English word "rescind", and which has become "scinder" in French still meaning something like 'to divide'), and also 'scire' ("to know" < "to split/distinguish truth from falsehood"). In Latin, 'scire' could be combined with the prefix 'ne-' ("not"), to give 'nescire' ("to not know about", "be ignorant of"), leading to the adjective 'nescius' ("ignorant"), which was the word eventually borrowed into English and which then gradually changed its meaning.

So if your notion of "original" is "originally appeared as a word that was ostensibly 'English'", then the meaning was probably close to 'ignorant', 'foolish'. But the origin of the word can be traced back much further than that. But then, there's nothing hugely magical linguistically speaking about Proto-Indo-European (or Latin, for that matter)--they're just arbitrary points (or rather, vaguely-bounded time periods) in history when we're able to posit something about the form of language that existed. Presumably the Indo-European form "skei" that we believe existed around 6,000 years ago itself developed from something, but we can't say exactly what. So in a strict sense, we can't say that this or anything else was the "original" form/meaning.

This is why arguments of the "well, this word really means X because that was the original meaning" type are generally spurious.

As to why it changed its meaning: well, in a sense, this word hasn't done anything remarkable. It's the normal state of affairs for words to change their meanings over time, and largely the more remarkable cases are words that have (e.g. because they're highly technical words usually consulted alongside their original, arbitrary intended interpretation whenever they are used) retained their 'original' interpretation.

Interesting.

An entry on amelioration (which is the upgrading or elevation of a word's meaning, as when a word with a negative sense develops a positive one.) specifically mentions nice as an example that has changed meaning since its original

•"The word nice is a classic example of amelioration . . .. This is a rare occurrence, compared with the opposite process of pejoration, or downgrading.

"The meaning of nice when it first appeared in Middle English (about 1300) was '(of persons or their actions) foolish, silly, simple; ignorant, senseless, absurd.'

" . . . A shift away from disparagement began in the 1500s, with such meanings as 'requiring or involving great precision or accuracy.' . . .

"The movement toward amelioration reached its apex in the 1800s with such meanings as 'kind and considerate, friendly.'" (Sol Steinmetz, Semantic Antics: How and Why Words Change Meanings. Random House, 2008)

This is what the NOAD says:

ORIGIN Middle English (in the sense [stupid] ): from Old French, from Latin nescius ‘ignorant,’ from nescire ‘not know.’
Other early senses included [coy, reserved,] giving rise to [fastidious, scrupulous] : this led both to the sense [fine, subtle] (regarded by some as the “correct” sense), and to the main current senses.

But the OED says that the meanings transformation is not very clear:

The precise development of the very divergent senses which this word has acquired in English is not altogether clear. In many examples from the 16th and 17th centuries it is difficult to say in what particular sense the writer intended it to be taken.