This answer is merely a summary of Stephen Gorard (2005), "Revisiting a 90-Year-Old Debate: The Advantages of the Mean Deviation" (PDF), which argues that we should abandon the standard deviation (SD) in favor of the mean deviation (MD).
The apparent superiority of SD [standard deviation] is not as clearly settled as is usually portrayed in texts
This "apparent superiority" stems from the following:
(1) SD is more efficient than MD under ideal circumstances
(2) it is easier to manipulate algebraically
(3) SD has now become a tradition, and much of the rest of the theory of statistical analysis rests on it
Regarding (1), Eddington (1914) had first shown that MD was empirically superior to SD. Fisher (1920) then countered that
that SD was more efficient than MD under ideal circumstances, and many commentators now accept that Fisher provided a complete defence of the use of SD.
The mean deviation is actually more efficient than the standard deviation in the realistic situation where some of the measurements are in error, more efficient for distributions other than perfect normal, closely related to a number of other useful analytical techniques, and easier to understand.
Another argument against SD is that
The act of squaring makes each unit of distance from the mean exponentially (rather than additively) greater, and the act of square-rooting the sum of squares does not completely eliminate this bias. ... our use of SD rather than MD forms part of the pressure on analysts to ignore any extreme values
Another proponent of the MD over the SD is Nassim Nicholas Taleb (2014), who claims that
It is all due to a historical accident: in 1893, the great Karl Pearson introduced the term "standard deviation" for what had been known as "root mean square error". The confusion started then: people thought it meant mean deviation. The idea stuck: every time a newspaper has attempted to clarify the concept of market "volatility", it defined it verbally as mean deviation yet produced the numerical measure of the (higher) standard deviation.