The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean

• Homotopy groups stabilize after taking suspensions (Freudenthal suspension theorem), or
• Cofiber sequences are fiber sequences (e.g., you've inverted the loops-suspension functors).

Does anybody know the first instances in which the words "stable" or "stability" were used to describe these phenomena?

As far as I can tell:

In the 90's and onward, Hovey, Schwede-Shipley, and Lurie, have used the second meaning to define stable (model, oo-) categories.

In the 60's, Adams and Boardman refer to stability in (the titles of) their books. I feel like, by this time, it was common to use the word "stability" in both ways (though I wouldn't know, not having been present).

In 1938, Freudenthal observed the first phenomenon. I would venture to say this might have been the beginning of "stable" phenomena in homotopy theory, but I don't know enough German to see if he even used the word "stable" in his paper.