Wikipedia has an acceptable definition of covariate, which matches exactly the Dictionary of Epidemiology. According to Webster, the first known use was in 1965. But the Google n-gram viewer has it showing up earlier, although I suspect many of the early documents are erroneously dated.
None of this really answers my question, though, which is: where does this now-ubiquitous term come from?
(Making comment into an answer.)
A good starting point is the list of earliest known uses of mathematical terms: jeff560.tripod.com/c.html which gives a reference to 1949 (but with covariate meaning specifically the "x" variable in a regression.) The same source says that the word variate goes back to Pearson in 1909, meaning a random variable.
COVARIATE. When R. A. Fisher introduced the analysis of covariance he called the regression variable the concomitant variable. In the 1940s the term covariate came into use for the variable in that role: see e.g. Biometrics, 5, (1949), p. 73. (JSTOR search) More recently covariate has been detached from the analysis of covariance (and from the analysis of experiments) to be used more broadly. It is now employed where independent variable or exogenous variable or regressor might also be used. An early instance of this usage is found in D. A. Sprott and John D. Kalbfleisch "Examples of Likelihoods and Comparison with Point Estimates and Large Sample Approximations," Journal of the American Statistical Association, 64, (1969), p. 477. (JSTOR search)
See the entries COVARIANCE, DEPENDENT/INDEPENDENT VARIABLE, ENDOGENOUS/EXOGENOUS VARIABLE, REGRESSION.
