Why are distributions "tempered"? Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term declines.
Schwartz himself seems to have used this term for the first time in his paper of 1951, Analyse et synthèse harmoniques dans les espaces de distributions,
Canad. J. Math. 3 (1951), 503–512, doi:10.4153/CJM-1951-051-5, and the French term was "distributions tempérées".
Google translates "tempérées" as "temperate".
I am not a native English speaker, but I understand "temperate" as a synonym of "moderate", which makes sense to me as a name of those distributions. For example, we say "temperate climate", not "tempered climate".
While "tempered" seems to be related to metallurgy, namely to a process making steel hard. What does tempering of steel have to do with distributions?
Why does the name "tempered" win in English?
 A: I think Michael, in the comment above, gave a convincing explanation for when to use "temperate" vs. "tempered". The answer by Matt also consolidates this point of "tempered" being associated to the result of an external action, since in all three examples the word "tempered" is followed by "by". Now where I disagree is about the conclusion to be drawn from this.
The delta function $\delta(x)$ is temperate because that's what it is.
It is not like it originally existed in the form $\delta(x)+e^x$, and then some deus ex machina came and tampered with it (sorry couldn't resist) by subtracting the exponential, and finally made it into a tempered distribution.
A: "Tempered" is used not just with steel but with thoughts and emotions:

*

*"What heaven can be more real than to retain the spirit-world of childhood, tempered and balanced by knowledge and common-sense." (Beatrix Potter, 1896)


*"Nationalism is power-hunger tempered by self-deception." (George Orwell, 1945)


*"Regret for the things we did can be tempered by time; it is regret for the things we did not do that is inconsolable." (Sydney Harris, 1951)
Perhaps that usage is what inspired the original mathematical users.
A: 
Can someone explain, why in English the name "tempered" wins?

Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p. 188 of
Schwartz, Laurent, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Internation Series in Mathematics. Paris: Hermann & Cie.; Reading, Mass. etc.: Addison-Wesley Publishing Company. 357 p. (1966). ZBL0151.34001:

A distribution $\mathrm T$ (that is to say a continuous linear form on $\mathscr D$) is termed
a tempered distribution if it may be extended to a continuous linear form on $\mathscr S$.

The usage was apparently already well-established by 1956 when G. Mackey reviewed a French paper of F. Bruhat and wrote (first occurrence of the term in MathSciNet):

The representations concerned are assumed to yield “tempered representations” when restricted to the Abelian normal subgroup being studied. Here tempered means being not too badly unbounded in a precise sense suggested by Schwartz’s definition of tempered distribution.

A: I've always understood tempering as a process to enable something to be more easily molded into something useful--to make it more malleable and robust--in this case the Fourier transform. As nLab puts it: The main property is that the Fourier transform of a TD is well-defined, and is itself a TD; and that it naturally extends the standard FT. This makes TDs the natural setting for solving (linear) PDEs.
Consistently, in metallurgy, tempering increases the ductility of a material, i.e., "the degree to which a material can sustain plastic deformation under tensile stress before failure," making it more useful.
Similarly, to temper one's emotions is to guide them, shape them, mold them, into productive channels, or at least less destructive / disruptive ones.
'To temper' has a long history. From Oxford Languages:
In Latin, temparare--to restrain, moderate $\to$ Old English, temprian, and Old French, temprer--to temper, moderate.
Old English temprian ‘bring something into the required condition by mixing it with something else’, from Latin temperare ‘mingle, restrain’. Sense development was probably influenced by Old French temprer ‘to temper, moderate’. The noun originally denoted a proportionate mixture of elements or qualities, also the combination of the four bodily humors, believed in medieval times to be the basis of temperament, hence temper (sense 1 of the noun) (late Middle English).
(I'm pretty sure Schwarz understood French.)
We use, in modern English, 'a temperate climate' to mean a moderate climate between tropical and harsh northern climates--a comfortable mixture of the two. We can say, "The climate of the coastal regions is moderated by the cool waters of the Pacific," but never, "The climate of ... is temperated by ...," rather, "The climate of ... is tempered by ... ." Temperate is purely an adjective like 'mild' whereas tempered is a verb (past participle) used as an adjective meaning to have been tempered--same grammatically as burned in 'a burned / burnt car'. Even the pronunciation of moderate changes according to whether it is being used as an adjective or as a verb or the past participle adjective moderated.
(Maybe a native French speaker can comment on parallels in the grammar in French.)
From all the considerations above, I see the delta 'function' as an construct of Heaviside and Dirac that has been molded into one amenable to the tastes of purist mathematicians (It's NOT a function!--the shrill mantra)--a morphing not really necessary for pragmatic physicists and engineers--by Schwarz and his theory of distributions. In that sense the delta function as a distribution that has been tempered by Schwarz and his theory seems fitting, just as the climate of coastal southern California is tempered or moderated by the Pacific waters. Of course one is free to say the climate along the coast is temperate, but that doesn't stress an agent and action that results in that quality.
To end with the Bard:
1591, William Shakespeare, Henry VI, Part 1, Act II, Scene 4, [22]:
Between two blades, which bears the better temper: […]
I have perhaps some shallow spirit of judgement;
But in these nice sharp quillets of the law,
Good faith, I am no wiser than a daw.
A: For me it's psychological: these are the distributions that are (well-)behaved, or (well-)tempered as opposed to the ill-tempered ones; "temperate" wouldn't work in "well-temperate" or "ill-temperate".
