The English word "Eve" is not the original name of the first human female. Her name was חַוָּה (chava) in Hebrew, which has a root connection with a verb לִחְיוֹת (lichyot) “to live,” as well as words such as חַי (chai) and חַיִּים (chayim), words for “life.”
וַיִּקְרָא הָאָדָם שֵׁם אִשְׁתּוֹ חַוָּה כִּי הִוא הָיְתָה אֵם כָּל־חָי׃
The human named his wife Eve because she was the mother of all living. (Gen 3:20)
In Hebrew, therefore, it makes perfect sense to call Adam’s woman חַוָּה (chava), because she will become the mother of “all living” כָּל־חָי (kol chai).
God notes that Adam preferred the voice of God’s beautiful and gracious creation (Eve) over the voice of his Creator, despite the explicit commandment not to eat of the forbidden tree. However, in great mercy to humanity, God curses neither Adam nor Eve, but the ground, saying: אֲרוּרָה הָאֲדָמָה (arurah haadama) "Cursed is the ground."
To Adam He said, “Because you did as your wife said and ate of the tree about which I commanded you, ‘You shall not eat of it,’ Cursed be the ground because of you; By toil shall you eat of it All the days of your life… (Gen 3:17)
The curse on the ground and its consequences on the humanity will later be lessened because of the faithfulness of 10th son of Adam, Noah, whose Hebrew name is associated with the word for "comfort" (Gen 5:29). This curse of the ground will be completely removed only in the last Adam, the son of Eve: Jesus (Yeshua), whose Hebrew name means "salvation."