What does it mean that John's Gospel does not have an account of the institution of the Eucharist?

The Last Supper, the institution of the Eucharist in the synoptic gospels, was the celebration of the seder feast at the commencement of the Jewish day of the Passover. Mark chapter 14 provides brief details of the preparation made for the feast, to be held in a large upper room that was already prepared for the occasion (Mark 14:14-15), and these details are copied in Matthew and Luke, but not John. The Jewish day begins at sundown, so Jesus was crucified on the day of the Passover.

In John's Gospel, Jesus was crucified on the day before the Passover:

John 19:14: And it was the preparation of the passover, and about the sixth hour: and he [Pilate] saith unto the Jews, Behold your King!

Because of this, Jesus could not celebrate the Passover feast on the evening before the crucifixion, in John's Gospel. Instead he and the disciples had their supper and then Jesus washed the feet of his disciples, providing a climactic even in lieu of the Last Supper. Ian Wilson says, in Jesus, page 120, that John, in his desire to represent Jesus as the new 'paschal lamb', wanted to have the crucifixion occur when lambs would have been slaughtered in the Temple in preparation for the Passover.

John spends several chapters, a relatively huge portion of his gospel in comparison to the other Gospels, recounting the events surrounding the last supper. Yet, he doesn't mention the Eucharist -- which had become very important in the church by the time he wrote but which, we know from Paul's writings, had been abused.
Perhaps he is using a literary device. Perhaps he knew that his readers would note the omission and look at what he wrote about instead, as a means of clarifying what communion is supposed to be about.
Take a look in the chapters about the last supper at what he talks about that the other gospels do not mention in their account of the same event. Ask whether those things are what communion should really be.

One theory is that John assumes his audience know one or more of the synoptic traditions and wrote to supplement the other gospels. If true, the "I am the Bread of Life" found in John 6:35 provides the philosophical background for the Eucharist custom. This might explain the reaction Jesus gets from His teaching:

The Jews then disputed among themselves, saying, “How can this man give us his flesh to eat?” So Jesus said to them, “Truly, truly, I say to you, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever feeds on my flesh and drinks my blood has eternal life, and I will raise him up on the last day. For my flesh is true food, and my blood is true drink. Whoever feeds on my flesh and drinks my blood abides in me, and I in him. As the living Father sent me, and I live because of the Father, so whoever feeds on me, he also will live because of me. This is the bread that came down from heaven, not like the bread the fathers ate, and died. Whoever feeds on this bread will live forever.” Jesus said these things in the synagogue, as he taught at Capernaum.—John 6:52-59 (ESV)

Even if you don't assume John wrote after the Synoptics, he would have known that his readers had access to the circumstances behind communion:

For I received from the Lord what I also delivered to you, that the Lord Jesus on the night when he was betrayed took bread, and when he had given thanks, he broke it, and said, “This is my body which is for you. Do this in remembrance of me.” In the same way also he took the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” For as often as you eat this bread and drink the cup, you proclaim the Lord's death until he comes.—1^st Corinthians 11:23-26 (ESV)

So instead of linking the practice to the final hours of Jesus' ministry, John links them to its start. He also connects the concept of Eucharist with the literal feeding of 5,000 in the first half of John 6.

Summary

John does mention the Eucharist, but not in the place we expect it to be from Paul's writing and the other gospels accounts. He ordered his account more for thematic reasons than for chronological or historical purposes.

I'm still looking into this next claim, but perhaps, while the Synoptic Gospels give us parables (think: "The Kingdom of God is like..."), John does not. In the Synoptics, it seems that Jesus says that he speaks in parables to the crowds, but reveals the secrets to the disciples. In John, he speaks more openly. What I think is that John contains the meaning of the parables found in Matthew, Mark, and Luke. And what does this mean in regard to the Eucharist? In John, Jesus institutes the Great Commandment: Love one another...as I have loved you. It is more than just "love your neighbor as yourself."

And the model for this love is that:

1) the Word became flesh (in the beginning of John), and a member of our fallen race. 2) very concretely, the foot washing, 3) the death and resurrection.

"Foot washing", serving our fellow man, is the only way we can emulate Christ on earth, unless we are in an exceptionally situation (think soldiers or firemen, etc) who die to save others.

To get to the point: perhaps the Eucharist celebration is the outward sign and symbol of that great truth, a mystical act that reminds us of our great commandment. So where parable is expressed in Matthew, Mark, and Luke, the meaning of the parable is explained in John. The true meaning of Jesus' life is represented by the bread and wine in those Gospels, but openly stated in John.

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John's Gospel does not contain many of the things recorded in the Synoptics, including the Sermon on the Mount, the Transfiguration, the virgin birth, the Great Commission, and the Ascension. In fact, the only miracle outside of the Resurrection that appears in both the Synoptic gospels and John's gospel is the feeding of the 5,000.

This doesn't mean that John considered these things as unimportant at all. If John were, in fact, written much later than the other gospels, then perhaps John did not feel the need to replicate material that was already well established.

John writes for a very specific purpose, asserting that Jesus is the Christ, the Son of God, and that by believing we may have life in His name. (John 20:30-31) So, the material he selects is done so for the purpose of supporting that premise. All the miracles and all the teachings center around that premise. The Eucharist appears to have not fit into that selection criteria for him, since it doesn't appear in the book.