Why did Jesus protest before performing his first miracle?

This is one of the reasons Catholics pray to Mary for intercession:

Our Lady of Lourdes, Mother of Christ, you had influence with your divine son while upon earth. You have the same influence now in Heaven. Pray for us; obtain for us from your Divine Son our special requests if it be the Divine Will.

From Our Lady of Lourdes novena prayer

Mary is interceding on behalf of the couple getting married.

The other half of the equation is the 'hour'.

Here's the other references to the hour coming in John's gospel:

4:21, 4:23, 5:28, 7:30, 8:20, 12:23, 12:27, 13:1, 16:2, 16:4, 16:25, 16:32, 17:1

NAB intratext

This is the beginning of Jesus's earthly ministry and in acknowledging that His hour had not yet come. Jesus states that His hour is indeed coming!

If Mary, whose heart would be pierced at the Crucifixion, knows anything that Jesus knows about the fulfilment of Jesus's mission on earth, then she knows that her next line, "do whatever he tells you to" is an acceptance that her Son's ministry is about to start (and conclude). Jesus may be giving Mary an opportunity to put off the beginning of his work, which she courageously forgoes.

Note, I just made the last part of this up, but I do pray the rosary and mediate on the Wedding Feast at Cana every Thursday so it's not that I haven't given it any thought

People debate this endlessly. It seems to me that the plain reading is that in Jesus' "original" plan, it was not yet time for him to reveal his divine nature by performing a miracle. And yet, he modified his plan to accommodate the desires of a believing woman.

I've heard some Bible teachers say that this does not mean that Jesus altered his plans to satisfy Mary. But I don't see how else you can read it. That's the plain meaning of the words.

The lesson I take from this is that God modifies his plans in response to human requests, i.e. prayer. If this were not so, then prayer would be pointless. That is, if God devised a perfect plan independent of human desires, and he will not change it to accommodate human desires because it is perfect, then prayer would be a meaningless and pointless exercise.

Does this mean that God acts in ways that are less than perfect to accommodate people? Not really, because his plan is presumably perfect AFTER TAKING INTO ACCOUNT what people want. Of course it doesn't follow that God will alter his plans to give people whatever they ask for -- there have been plenty of discussions about when and why God grants some prayers and not others.

First and foremost, Jesus told the woman (mother) that His time had not yet come. Keep in mind this is kind of like the "Bruce Almighty" effect, God is not a God that only uses His powers to further Himself.

If there was somebody that needed to be healed, I'm sure that he would have done it without hesitation. But this miracle was only so that people could continue to party.

I believe Jesus is talking about His public ministry, not about any miracle. A closer reading of John 2:1-12 can bring out some possibilities as follows.

1. Mary knew about Jesus's power and His calling. (Events in His childhood described in Gospel of Mathew and Luke points to that).

2. She would have probably witnessed His power in events which happened in family circles which may not be recorded in Gospels (Inferred from John 21:25 - And there are also many other things that Jesus did, which if they were written one by one, I suppose that even the world itself could not contain the books that would be written).

3. This marriage party being a public event with probably many from the locality and around, Jesus did not want to do a public miracle since it was not yet time for His public ministry.

The above reasons would explain the answer Jesus give to His mother.

Now we can see Mary's response to those known in need (here servants who were handling store). She says to them to 'do whatever He says'. I find it consistent, since it appears now Mary understood that Jesus power cannot be controlled by any body else. We can see Jesus did the miracle in private where only servants, His disciples who were with him that time and probably Mary only knew about it. This can be inferred from verse 9 where it says master 'did not know where it came from'.

So in summary,

What did Jesus mean that his hour had not come?

Jesus means that the time of His public ministry not come.

Why do his actions seem to contradict his words?

His action did not contract His words. He did it in private without the group assembled there knowing it, not even the master of ceremonies. He ensured that it was known only to a very private circle (need to know basis in modern parlance). The verse 11 says, disciples believed in Him. Servants might not have known who He is and others did not even know about it.

Trust this helps us to understand this passage and Jesus's words better.

Most likely because he didn't want to gain a lot of notoriety so early on, and doing something like this, especially in such a public setting, would cause a lot of people to notice. But in the end, he did it, probably because it was his mother asking, and he was required by the Ten Commandments to honor his father and his mother.