Another interpretation: because the Perat rose up and encircled the whole land of Israel, of which it is written, "For what great nation is there" (Deuteronomy 4:7), it is called great. As the common proverb says: the servant of a king is a king; cleave to the official and they will bow to you.
Why the Euphrates Is Called Great for Encircling Israel
Yalkut Shimoni on Torah 21:5
דָּבָר אַחֵר, לְפִי שֶׁעָלָה פְּרָת וְהֵקִיף כָּל אֶרֶץ יִשְׂרָאֵל, דִּכְתִיב בֵּיהּ (דברים ד, ז) "כִּי מִי גּוֹי גָּדוֹל", נִקְרָא גָּדוֹל מָשָׁל הֶדְיוֹט אוֹמֵר, עֶבֶד מֶלֶךְ מֶלֶךְ, הִדַּבֵּק לְשַׁחְוָר וְיִסְגְּדוּן לָךְ.