What innovation made WiFi amenable to mobile devices without prematurely depleting them of power?
In June 2004, the Economist,

A financial / economy magazine should be read with a lot of caution when it comes to technical claims made therein – the people writing, interviewing, editing such a magazine aren't engineers!
Just the same applies the other way around: I wouldn't trust IEEE Spectrum to be the best source for discussion of economy.

in reference to the nascent WiFi standard

&quot;Nascent&quot; as in &quot;standardized five years before&quot;. If you consider the speed of mobile phone development that we already saw in the 2000s, that isn't &quot;nascent&quot;. That's &quot;mature, if not a little old already&quot;.

stated that &quot;It is currently too power-hungry for mobile devices&quot;

Well, either the Economist was simply wrong, or something fundamental happened between that article and the release of things like third-gen E-series Nokia phones a year later.
Of course, there was ongoing development in the area of batteries, making higher-power devices feasible. At the same time, power efficiency of silicon of course continuously increased.
But let's be honest here: the Economist was wrong.
Wifi integration was not yet market ready in 2004 for mobile mass products, but power demand was just one of many things, and I'd say that in 2004, companies actually designing phone SoCs would already have had feasibility studies done that showed it was very realistic to have a Wifi phone out a year or two later. Development of such things don't happen in a couple of months. It takes planning and severe investments.

So how did WiFi go from being extremely power-hungry to being suitable to mobile device usage?

It was never &quot;extremely power-hungry&quot;: That's not a true claim. As it was standardized, it was meant to later be implemented in mobile devices. That was among the many goals standardization committees have. So, the whole standard was written by people who had a good idea that it has to be possible to implement it efficiently enough to put it in a mobile device later on.
Andy put it very well: the reason for it then becoming actually feasible is called progress.
The same as for every other technology:

increase in battery capacity
improvement in semiconductor static and dynamic power losses
improvement in the efficiency of power supplies
improvement of algorithms

in particular for Wifi:

lower-power oscillators
better ADCs
smarter synchronization algorithms being available
higher-efficiency implementation of &quot;old&quot; forward error correction methods.

Oh, by the way: just found out that Nokia made the Wifi-connected Nokia 9500 Communicator, and put it to market 6 months after the Economist claimed it was too power-hungry. Sure, that was slow Wifi (802.11b, 11 Mb/s max) by modern standards, but a year after that they brought out the 802.11g (54 Mb/s) 9300i, and another year later, the whole E61, E7x series, and Wifi became a standard feature for smart phones at that point.