I'm not sure that there is any real profound significance. The names do have meanings as follows:
- Pishon: increase
- Gihon: bursting forth
- Tigris - rapid
- Euphrates - fruitfulness.
One thing that may be of note is that Adam may very well have given them these names. God specifically had Adam name the animals. This seems to be inherent in the dominion which God gave to man over all the earth. God Himself named things in Genesis 1, calling the light "day", the darkness "night", the expanse "heaven", the dry land "earth", and the waters "seas". (Genesis 1:5, 8, 10) In Genesis 2, it is the man who gives names to things:
Out of the ground the Lord God formed every beast of the field and every bird of the sky, and brought them to the man to see what he would call them; and whatever the man called a living creature, that was its name. Genesis 2:19 NASB
So, while we are not specifically told that man named the rivers, it is certainly possible. If so, it would be another demonstration of man's dominion over the earth.
Furthermore, the very fact that specific rivers are named and described with specific details strongly suggests that the account is not merely a story but an actual historical event and that Adam was, indeed, an actual historical figure. The genealogy of Adam to Abraham, Moses and David is similar in that fictitious characters cannot trace their lineage to real people.
If the account were, in fact, just a story of creation, the fact that one of the rivers flowed around the entire land of a certain region would be quite odd, as would be the mentioning of gold, bdellium and onyx stones being present in a particular region. The story is written as if it were a real place where real events occurred and where people could visit if they so chose.
In conclusion, the significance would be to possibly illustrate mankind's dominion over the earth and also that the story is actually a historical event.