In answer to your first question, the Hebrew Bible provides several "etymologies" of proper names. For example, the name "Israel" is said to derive from the Hebrew for "struggle" (Genesis 22:29).

These "etymologies" don't come anywhere near modern standards of linguistic rigor, they may well be incorrect, and there may well be earlier attempts at "etymologizing" out there. But I hope this helps.

In answer to the second part of your question When was it first recognized that multiple languages may have diverged from an earlier language?

Many languages have obvious connections. For example, Latin and Greek share many similarities. However, the ancient Greek grammarion Tyrannion explained this by claiming that Latin was derived from Greek. One of the earliest claims that many languages were derived from a now-extinct ancestor was by William Jones, a British colonial administrator in India, who famously claimed in 1786:

The Sanscrit language, whatever be its antiquity, is of a wonderful structure; more perfect than the Greek, more copious than the Latin, and more exquisitely refined than either, yet bearing to both of them a stronger affinity, both in the roots of verbs and the forms of grammar, than could possibly have been produced by accident; so strong indeed, that no philologer could examine them all three, without believing them to have sprung from some common source, which, perhaps, no longer exists

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