Summary: Jesus' thirst on the cross was a product of His humanity and His deep spiritual thirst for the Living Water He was making available to us.
Jesus is done. He's done everything He came to do. He hasn't declared it yet, but, in fact, his work on the cross is an accomplished fact and he is ready to die. However, there is one last thing He wants to round out, one last detail.
He is filling up the Scriptures. This is not an explicit prophecy that must be addressed, not a "t" to be crossed or an "I" to be dotted ... it is a nuance with which Jesus wants to shade this event. He wants to take a scripture and endow it with a deeper, richer texture. In so doing, Jesus is taking this last action of his life, this last expression of His humanity and charging it with something more:
Later, knowing that everything had now been finished, and so that Scripture would be fulfilled, Jesus said, "I am thirsty." A jar of wine vinegar was there, so they soaked a sponge in it, put the sponge on a stalk of the hyssop plant, and lifted it to Jesus' lips. (John 19:28-29 (quickview)  TNIV)
Probably, no other statement of Jesus on the cross so forcefully expresses His humanity
His care for His mother is human, surely, but it shows a divine insight and a transcendent love. This statement, on the other hand, may be read as something straight out of any man's medical chart. It could be the cry of the lowest criminal or sinner. I love the way Charles Spurgeon expressed it in his message, "The Shortest of the Seven Cries":
“The sea is his, and he made it,” and all fountains and springs are of his digging. He poureth out the streams that run among the hills, the torrents which rush adown the mountains, and the flowing rivers which enrich the plains. One would have said, If he were thirsty he would not tell us, for all the clouds and rains would be glad to refresh his brow, and the brooks and streams would joyously flow at his feet. And yet, though he was Lord of all he had so fully taken upon himself the form of a servant and was so perfectly made in the likeness of sinful flesh, that he cried with fainting voice, “I thirst.” How truly man he is … (Spurgeon The Shortest of the Seven Cries).
As truly God and truly man, this cry is a cry of true humanity, for humans cannot long endure thirst.
When we feel our last fever. When our bodies rebel and bring on an overwhelming sweat that drains us of our moisture. When our throats tighten and we are left unable to swallow. When we have bled, we too will thirst. Our bodies will cry out for the moisture they need to thrive.
For Jesus, then, this is not news. The Man has lost a lot of sweat, blood and tears. Dehydration is, after asphyxiation, according the Journal of the American Medical Association, one of the secondary causes of death by crucifixion.
He's already been offered something to drink a couple times. The first time, early in His ordeal, it was mixed with myrrh or gall. This was a pain killer, specially prepared by the Jews as a compassionate act for people being crucified, to ease their suffering. Jesus spits it out. He wants none of it. He will drink the whole cup of suffering. He will consciously and willfully endure the pain He has chosen to embrace for your eternal well-being and mine.