Who "invented" the standard error of the mean? I need the earliest available source. I already searched statistic books of Andy Field, Bortz & Schuster, Rasch & Friese, Wikipedia, Google and I asked 3 colleagues who teach statistics.
 A: Like many ideas in the history of statistics, it's not as simple as that; the standard error of the mean doesn't burst sui generis from a single mind. The term is only about a century old but the underlying concepts are considerably older.
It seems as if the term "standard error" was first used by Yule in 1897 in relation to the standard error of a residual. This is only 3 years after the first use of the term standard deviation!
Yule then used the term "standard error" in relation to other statistics in his 1911 book. However the concept of the standard error of the mean already existed at that point (without the name); for example Student was discussing estimating it using the sample-based estimate $s/\sqrt{n}$ three years earlier. However actually dividing $s$ by $\sqrt{n}$ is considerably older than that - much older than the term standard deviation (though the concept was much older) -- but previously $s/\sqrt{n}$ was generally multiplied by a constant ($\Phi^{-1}(\frac34)\approx 0.67448975$, which converts $s$ to an estimate of the mean absolute deviation from the mean for a normal population) and the result was called the probable error of the mean. In that context, the use of $s/\sqrt{n}$ dates back at least to 1853.
So the notion of measuring uncertainty in the estimate of the mean already existed at least 40 years prior, and the $s/\sqrt{n}$ calculation itself was already there. Essentially as soon as statisticians moved to focus more on standard deviation rather than mean deviation as their primary measure of spread, they simply dropped multiplying "s/\sqrt{n}$ by "0.67448975" from the calculation of the probable error of the mean, but it was already a calculation in long use.
References:
Student, (1908),
"The Probable Error of a Mean"
Biometrika, Volume 6, Issue 1, March, Pages 1–25,
Earliest Known Uses of
Some of the Words of Mathematics
(several entries by John Aldrich)
https://mathshistory.st-andrews.ac.uk/Miller/mathword/
A: Just by using some logic:
The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean. See this Wikipedia page for more details.
Suppose you are talking about the standard error of the mean in the context of normal distribution. If that is the case, then you would not go entirely wrong by even citing Gauss (1809) when he first introduced the normal distribution in his monograph. This is because normal distribution is defined by two parameters: the mean and variance. The square root of the variance is the standard deviation. And, when we are talking about the distribution of sample means in the population, the standard deviation is the standard error. 
Hence, I would use the following citation:

Gauss, C. F.(1809). Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm [Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections] (in Latin).


Important edit:

If you want to be extra technical when it comes to attributing the notion of normal distribution to Gauss, you may want to say a couple of sentences about De Moivre (1738) and his book the Doctrine of Chances. Some believe that he should be credited with the introduction of normal distribution rather than Gauss (Johnson, Kotz & Balakrishnan. 1994; Le Cam & Lo Yang, 2000). However, this history subsection of the Wikipedia page on the concept of normal distribution provides some compelling reasons challenging such an alternative view. One of the main arguments was that De Moivre lacked the notion of probability density function.


References: 
De Moivre, A. (1738). The Doctrine of Chances.
Gauss, C. F.(1809). Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm [Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections] (in Latin)
Johnson, N. L., Kotz, S., Balakrishnan, N. (1994). Continuous Univariate Distributions. Wiley.
Le Cam, L., & Lo Yang, G. (2000) Asymptotics in Statistics: Some Basic Concepts (second ed.). Springer.
