I looked around for a bit, I couldn’t really find anything on word meaning, but for a Christian is a different story. Being a Christian myself I know full well CS Lewis and his beliefs. Lewis wrote Narnia with a lot of different Christian themes as did his friend Tolkien in LoTR. Like Aslan dying and coming back has certain connotations. There are other references that would probably take up a few pages but you also must understand is that this is also fantasy, while there are definitely real world connotations and meanings, Lewis wrote this to be a great and fun story.

Narnia is the ancient name of Narni - a town about half way between Assisi and Rome. 

CS Lewis may have gotten the name Narnia from a town in Italy. There was a legend of "Blessed Lucy of Narnia" (it may have been the inspiration of Lucy Pevensie, the first child of the four children to discover Narnia).

As for the Kingdom for Christians, some arguments could be made. CS Lewis never intended Narnia to be allegory. He actually said this-

In the Focus On The Family Radio Theatre, in the prologue of The Last Battle, Douglas Gresham states-

"Where The Lion, The Witch, And The Wardrobe gives a promise, The Last Battle sees that promised fulfilled."

You could argue that the Chronological order of the Chronicles is like Genesis-Revelation. Edmund is Judas and Barabbas (Edmund was supposed to die on the stone table, like Barabbas was supposed to be the one crucified); Eustace is Paul (Saul going blind and Eustace turning into a dragon made them realize how they've mistreated others); Lucy has the faith of a child, being the youngest of four children; Peter is the apostle Peter (Peter Pevensie was chosen to be the High King, like the Apostle Peter was chosen to be leader of the church); Aravis and Emeth are represented of how God accepts all people, no matter their racial backgrounds. Aravis could be like Rahab and Ruth, who were both Gentiles and accepted into the Jewish family; Emeth has often be viewed as the Ethiopian Eunuch in Acts.

@jasminetarkheena  you could, but it would be wrong. You would be closer if you said they reminded us of those Bible characters. 

@coracle That, too.

I think that in The Last Battle, where Shift would say, "This is what Aslan wants you to do", it's kind of like if we say "This is what God wants me to do" or "It is God's will", but in reality, it's not the case.

I suppose that Rishda and Emeth could be viewed as the Pharisee and the Tax Collector- Rishda demonstrates arrogance, like the Pharisee, and Emeth demonstrates humility, like the Tax Collector.

As other's have discussed, the series does have references to the Judeo-Christian Bible that C.S. Lewis believed in.

What is the meaning of this Kingdom for a Christian?

I too am a Christian, and would suggest looking into the Bible itself for an explanation as to what the Kingdom is. In the years that Jesus walked the earth, he often said he was here to bring the "Kingdom of God" or "Kingdom of Heaven." In the New Testament, we see these two mentions of the Kingdom over 80 times. The primary books they're found in are the Gospels of Matthew, Mark and Luke.

I think you have to be careful about making direct corollaries to anything in the Christian Bible because Lewis shaped a lot of what he wrote in the Chronicles from the things other than it.  Lewis himself warned people that you can't make it into strict allegory.  That's because a lot of what is written in the Chronicles isn't strictly from Christian theology.   Lewis was a scholar who was well aware of other myths, legends, and spiritual practices and drew from them.  There is also a healthy dose of psychology and an understanding of children in it.    

I've always felt that Narnia provided a place where the Pevensies could freely explore what was within them as people and to learn about themselves.   They were given trials where they could see what they were able to do, and their opinions would be heard by adults and taken into account.   It was a place were they could experience the adult world in a way that was less threatening than it was on Earth.  Even when I read this as a child I remember thinking how cool it was that children near my own age where given the opportunity to do adult things and have support from beings like Aslan in doing them.