Your Protestant friend is missing a lot, if he thinks “I thirst” was just something random that Jesus happened to say. Protestant as well as Catholic Biblical commentaries have always regarded “I thirst” as a particularly meaningful statement.
First off, it was important for Jesus to be offered “wine and gall” as part of fulfilling Biblical prophecy about the Messiah. (Ps. 68:22/69:21 – “They gave me gall… and ijn my thirst, they gave me vinegar to drink.”) The sponge with wine/vinegar and gall also included “hyssop” (John 19:29), which goes to the way the Passover blood of the lamb was placed upon the door of the house (and Jesus is the Door as well as the Lamb!); and was on a “reed,” which goes to “The bruised reed he shall not break.” (Is. 42:3/Mt. 12:20).
Second, it tied into what He said - promised - at the Last Supper: “I will not drink wine again until I drink it new in the Kingdom of God.” When He said, “I thirst,” He was anticipating eagerly His inauguration of the Kingdom of God by dying and rising, when He would drink “new wine.”
(And very possibly it ties into the “fourth cup of Passover” theory.)
Finally, “I thirst” is often seen as Jesus’ longing to have the Cross done with, so that He could have people redeemed. He hungers and thirsts for justice; He hungers and thirsts to save our souls and make us happy with Him forever.
Mother Teresa based her whole ministry to the poor on her meditations on Jesus saying, “I thirst.” It’s the foundation of her religious order’s spirituality.
So yeah, that’s a Big Deal quote from Jesus. I’ve hardly dug into it at all.