What is the etymology of "regression" as in finding the coefficients of polynomials?
Latin "re-" ("back") plus "-gredior, -gredi, -gressus sum" ("go"); the "-ion" suffix is common for forming nouns.
Thus "regression" literally means "going back". It is more commonly used in a figurative sense (as the opposite of "development"). The mathematical sense you mention comes from the idea that one would normally use a formula to calculate coordinates of a curve, but in "regression" one is starting with the coordinates and "going back" to the formula.
I think it inherits it merely from being part of the larger concept of regression analysis. From that, as per Wikipedia:
The term "regression" was coined by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean). For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context.