A great example of terrible product name(s) are Element, Mastodon and even Bluesky.
Frankly I'm tired of these un-google-able cutesy names that aren't unique. If you want some examples look at the other suggestions in this thread (Jeeves, Mother, Intuition). ChatGPT is only exactly one thing, and you "chat" with it, that's good enough to me.
"Bard" is alright but harder to uniquely google, and Bing's ChatGPT implementation is the terrible since I cannot even figure out what the name is ("Bing AI?").
Or “Intuition”.
But perfectly suitable to such a dorky product.