The last verse of Psalm 83 is not to be understood that that is the only Name of God, rather the correct translation is "Then they will know that You whose Name is [a four-letter name], are alone, most high over all the earth".
The answer to your friend is that we know what the Name is, but we are forbidden to pronounce it (as I will try to explain), and we don't even know how to pronounce it. It has not been pronounced in Judaism for more than 2,000 years, under very special conditions in the Temple in Jerusalem.
In general, I think it is correct to say that the "main name" of God is the Four-Letter one in that verse in Psalms and many places in the Torah. The Four-Letter Name indicates the concept that God is timeless and infinite, because the letters that comprise His name correspond to "He was, He is, He will be". It is also a Name associated with the attribute of mercy, with which He created the world.
This name is forbidden for us to pronounce in the same way as it is spelled. This Four-Letter Name should be pronounced "Adonai" (meaning my Master) in prayers, blessings or Torah readings, or otherwise one should refer to God as "Hashem" - which means "The Name" in Hebrew. This not saying the Name in any form without true need is a show of honor and respect for the Four-Letter Name since it avoids saying the Name unnecessarily.
There actually are many Names of God. In fact the Code of Jewish Law states that God has seven Main Names, and many other "sub-names" that bring the total to seventy. The number seventy represents totality. Jewish Tradition teaches that there are seventy main languages in the world, and there are seventy main peoples that populate the world.
A name is a person's identity. When we speak of the Names of God it is not possible to understand God's "identity". However, each single Name reveals another facet of God's Presence in this World.