Descartes was motivated by proto-coordinate constructions in Apollonius's Conica, who regularly drew parallel segments ("ordinates") to a diameter of a conic in his demonstrations. As for the word itself, it might go back to the terminology of Roman surveyors (lineae ordinatae), the early medieval occurrences are in rectae ordinatim applicatae (straight lines designated in order), but as a mathematical term it was coined by Leibniz some time between 1677 and 1692.
In an early edition of his History of Mathematics (1909), p. 216 Cajori boldly wrote "The Latin term for "ordinate" used by Descartes comes from the expression lineae ordinatae, employed by Roman surveyors for parallel lines". But in the later editions, e.g. the 1929 edition, p.175, he changed his mind:
"Here Descartes follows Apollonius who related the points of
a conic to the points of a diameter, by distances (ordinates) which make a constant angle with the diameter and are determined in length by the position of the point on the diameter. This constant angle is with Descartes usually a right angle... It is also noteworthy that Descartes does not formally introduce a second axis, our $y$-axis. Such formal introduction is found in G. Cramer's Introduction d l'andyse des lignes courbes algibrique, 1750; earlier publications by de Gua, L. Euler, W. Murdoch and others contain only occasional references to a $y$-axis. The words "abscissa", "ordinate" were not used by Descartes.
In the strictly technical sense of analytics as one of the coordinates of a point, the word "ordinate" was used by Leibniz in 1694, but in a less restricted sense such expressions as "ordinatim applicatae" occur much earlier in F. Commandinus and others. The technical use of "abscissa" is observed in the eighteenth century by C. Wolf and others. In the more general sense of a "distance" it was used earlier by B. Cavalieri in his Indivisibles, by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome, and by others. Leibniz introduced the word "coordinatae" in 1692."
Miller's Earliest Known Uses of Some of the Words of Mathematics gives the following earlier citations for Leibniz's uses:
"Leibniz used the phrase "per differentias ordinatarum" in a letter to Newton on June 21, 1677 (Scott, page 155).
Leibniz used the term ordinata in 1692 in Acta Eruditorum 11 (Struik, page 272).
For the word in English the OED has a passage from 1706: H. Ditton An Institution of Fluxions p. 31 “'Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse.”"
Struik's Source Book in Mathematics, 1200-1800 in a footnote to the cited page has the following, confirming some of Cajori's claims:
"Note the Latin term abscissa. This term, which was not new in Leibniz's day, was made by him into a standard term, as were so many other technical terms. In the article "De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata... ," Acta Eruditorum 11 (1692), 168-171 (Leibniz, Mathematische Schriften, Abth. 2, Band I (1858),
266-269), in which Leibniz discusses evolutes, he presents a collection of technical terms.
Here we find ordinata, evolutio, differentiare, parameter, differentiabilis, functio, and ordinata
and abscissa together designated as coordinatae. Here he also points out that ordinates may be given not only along straight but also along curved lines. The term ordinate is derived
from rectae ordinatim applicatae, "straight lines designated in order," such as parallel lines. The term functio appears in the sentence: "the tangent and some other functions depending
on it, such as perpendiculars from the axis conducted to the tangent.""
Scott in History of Mathematics quotes directly the cited Leibniz's letter to Newton, explaining the differential triangle:
"From this it is clear that to find the tangents is nothing more than to find the differences of the ordinates corresponding to given (equal) differences of the abscissae".