Eve's name in the above verse is translated from the Hebrew word חוּה
, or chavvâh, Strong's Concordance
#H2332 and literally means "life" or "living." Inserting the actual Hebrew meaning of Eve's name into Genesis 3:20 gives us:
"And Adam called his wife’s name LIFE, because she was the mother of ALL LIVING."
Eve was named LIFE, or the mother of ALL living, because out of her would spring the ENTIRE (not just some of or most of) human race! Eve's name does not leave any room for human beings to BEGIN to come into existence by any other woman or means (e.g. God creating more "Adams," or another set of humans like Adam and Eve, etc.)
Now let's answer your second question regarding how the human family could have gotten progressively bigger if only Adam and Eve were created.
The Bible mentions, by name, three children of Adam and Eve: Cain, Abel, and Seth. Cain was the firstborn. Seth was born sometime after Cain killed Abel, when Adam was one hundred and thirty (130) years old. Genesis 5 tells us that after Seth was born Adam and Eve had other children:
"And Adam lived one hundred and thirty years, and begot a son in his own likeness, after his image, and named him Seth. After he begot Seth, the days of Adam were eight hundred years; and he had SONS and DAUGHTERS." (Genesis 5:3-4)
Adam lived a total of nine hundred and thirty (930) years! It would be safe to assume that Eve lived at least as long, maybe even longer, than Adam.
Even if we assume Adam and Eve only had three children in their first 130 years together, they had PLENTY of time after this to take full advantage of God's blessing and obey His command to procreate:
"Then God blessed them (Adam and Eve), and God said to them, 'Be fruitful and MULTIPLY; FILL the earth and subdue it . . .' " (Genesis 1:28)
A footnote in the writings of the Jewish historian Flavius Josephus (Antiquities of the Jews, Book 1, Chapter 2) stated that a very old tradition said Adam had a total of THIRTY-THREE (33) sons and TWENTY-ONE (21) daughters!
Additionally, since Adam and Eve were created perfect, their genetic makeup would be perfect and so would that of their first few children. With no genetic problems to enter the picture, God allowed them (for a time) to intermarry and fairly quickly begin to populate the world. As time went by, however, genetic imperfections would make close-kin marriages impractical and a real risk not only for the mother but also the children who could more easily be born with genetic defects and handicaps.
Some mathematicians have calculated that at the time of Noah's Flood, which was roughly 1,600 years after the creation of Adam, that the population on earth could conservatively been at least ONE BILLION people!
The Bible record leads us to conclude that God personally created only two humans, and then let the biological processes and urges He placed in them do the rest to populate the earth.
Thank you for submitting your question about God creating a second Adam.