Why does the serpent tempt Eve? Because it can.
People sometimes ask why the serpent, specifically, had a motivation to cause man’s fall. But I think this is the wrong question. What the text actually tells us, the very first thing it tells us about the serpent, is this: “The serpent was the most cunning of all the beasts of the field.” This answers the question, Why could the serpent, specifically, cause man’s fall? Nowhere does the text ask why the serpent wanted to, because we’ve already been told that – in 1:28.
What the text tells us is that the serpent was unique in its capabilities; it does not say that the serpent was unique in its motives. I think the serpent’s motive was shared among all the animals: resentment towards man for man’s having been given dominion over all other life forms. (Steinsaltz on 3:1: "Perhaps it sought to usurp man's position atop the hierarchy in the Garden of Eden.")
All the animals had the motive, but only the serpent had the method; all had the intent, but the serpent alone had the capability.
This, then, is the first instance of envy and jealousy in the Bible, even before the well-known brothers whom we’ll meet in the next chapter. And it is also of the same theme: rather than wanting to better itself and improve its own standing, the serpent wants to bring the other guy down. This is the nature of envy and it’s an all too common human weakness.
Venturing just a little bit into symbol, we might take the snake – and, in my reading, the putative rebelliousness of the animal kingdom generally – as a metaphor for how our lower nature, our animal instincts, will often use rationalization to get us to do things we know we shouldn’t do.
“He named her Life [chava]” – as Steinsaltz drily observes, he could have called her a lot of things at that moment. But he didn’t. He named her Life.
“- because she was the mother of all life.” I think the verb here [hayetah] really wants to be translated as “had become” – “she had become the mother of all life.” And indeed that’s exactly how Steinsaltz reads the later verse, “the man had known his wife”.
So, what is really going on here? I think she must have been already pregnant, and perhaps she told the man her wonderful secret right then and there. And now, suddenly, the fruit, the fall, the curse – none of that matters now, because they are about to bring a new human life into the world.