Preface:

There has been much discussion in comments and other answers about how to interpret the term "Wi-Fi"; what it should or does mean by virtue of historical and common usage and implied meaning. There is no "right answer" to that. This answer can only address what the term is officially supposed to mean, and the historical background that has given rise to these arguments.

Wi-Fi did not originate as an abbreviation

The Wi-Fi name and logo were designed as simply a trademark. To quote the article Wi-Fi Definition is Not Wireless Fidelity on Webopedia,

Wi-Fi isn't short for anything

It was, however, a play on words with "hi-fi".

Background

The term Wi-Fi, commercially used at least as early as August 1999, was coined by brand-consulting firm Interbrand Corporation. The Wi-Fi Alliance had hired Interbrand to determine a name that was "a little catchier than 'IEEE 802.11b Direct Sequence'". Phil Belanger, a founding member of the Wi-Fi Alliance who presided over the selection of the name "Wi-Fi", also stated that Interbrand invented Wi-Fi as a play on words with hi-fi, and also created the Wi-Fi logo.

Source: https://en.wikipedia.org/wiki/Wi-Fi

If you look at the brand names Interbrand has created, most are meaningless sounds that are catchy to say, or nonsensical combinations of word pieces to create a new "word". The objective of a brand name is to conjure an association in the user's mind; the definition is the product. Interbrand's proposal of "Wi-Fi" was likely because it had a letter pattern reminiscent of "hi-fi" and rhymed with it, making it a good marketing "word".

As I'll describe shortly, the Alliance was trying to promote the use of wireless LANs to the home market to transfer audio and video. The similarity to "hi-fi" was a good fit, but not because "hi-fi" meant "high fidelity".

Most people know that "hi-fi" had been shortened from "high fidelity". However, "Wi-Fi" was coined half a century later, and "hi-fi had evolved. It was no longer short for, or synonymous with, "high fidelity". Its last common usage was as slang for ubiquitous, consumer-grade audio equipment or reproduction.

So the "link" between "Wi-Fi" and "hi-fi", at that point, was simply the association with music and multi-media, not the meaning of the syllables. Just because "hi-fi" was shortened from "high fidelity" doesn't reserve the "fi" in every word that imitates that letter pattern to meaning "fidelity".

Origin of "Wireless Fidelity"

After adopting the name and logo, some of the Alliance members had a problem with the concept that something that looked like an abbreviation didn't have a literal explanation. As a compromise, it was agreed to include, along with the name, the tag line, "The Standard for Wireless Fidelity". This implied a word association without there actually being one. As Phil Belanger describes it:

Wi-Fi doesn't stand for anything. It is not an acronym. There is no meaning. This tag line was invented after the fact. ... The tag line was invented by the initial six member board and it does not mean anything either. ... And "Wireless Fidelity" - what does that mean? Nothing. It was a clumsy attempt to come up with two words that matched Wi and Fi. That's it.

Excerpted from WiFi isn't short for "Wireless Fidelity"

Further explanation from Belanger:

The current confusion seems to stem from a brief period early in the days of the Wi-Fi Alliance when a regrettable tag line was added that stated, "The Standard for Wireless Fidelity." This was not part of the original name and was not created by Interbrand, but it was added as an afterthought in an attempt to help users make sense of the new and somewhat nonsensical word, "Wi-Fi." We weren't creating standards -- we were promoting an existing standard. One of the motivations was that we were trying to expand the use of WLANs to the home market, so this notion of "wireless fidelity", some people felt like if they're going to transfer audio and video around their house, then maybe that has some of the appeal. We have this name Wi-Fi. What two words have "wi" and "fi" starting them? Maybe it can help support our goal?" By the end of 2000, the pointless tagline was dropped and the term "wireless fidelity" was supposed to disappear into the ether. But somehow, as the Wi-Fi brand gained traction, so did the mistaken notion that it was "short for wireless fidelity."

Excerpted from 'Wireless Fidelity' Debunked

Conclusion

Officially, "Wi-Fi" has no meaning. The Wi-Fi Alliance created the impression that it stands for "wireless fidelity" and has spent the last 16 years trying to correct that. However, the word association is still well-entrenched and still repeated.

Addendum

The next generation replacement for Wi-Fi may be a technology currently in development, based on data transmission embedded in LED room lighting. As described in another SU post, the developer is among the people who believe "Wi-Fi" stands for "wireless fidelity". As a cute knock-off of a cute knock-off, he coined the name "Li-Fi", and explicitly called it "light fidelity".

So regardless of the Wi-Fi Alliance's original intent, "fidelity" may be here to stay. The decision to use the tag line created the "gift that keeps on giving".