Dear Babu George,
If it is your conviction that the city in Genesis 24:10 was the home of Nahor. No one denies that fact. That is why a city was called after a persons name, in the ancient times, in Genesis. At the same time, we should also remember that it was not called the city of Bethuel or Laban, who were also residence of that place.
Your conviction also retains the idea that there was no city with that name, which archeaology has shown otherwise.
Your logic could be considered as, where you live is the 'city of Babu George.' This may or may not be true. Before you began to live in Chennai, you might have lived in another place. In fact, there may be a series of places to mention as your place of residence, in the past. Then all of them are cities of Babu George. Not one city. If there is one city that could be called as the "city of Babu George," then it should be known after your name. [Note: This is only a logical argument.]
Now let me point your attention to Genesis 23:10 and 18. In these two verses, there is an expression, "... of his city." What is the meaning of that expression? How is it different from the expression, "city of Nahor?" There are more verses to study. For example: we read about a "city of Shechem," "city of Arbah," etc.
This is not an issue of doctrines, but of logical reasoning of man's mind. Unfortunately, the archaeology is not on your side and it has already indicated the existence of a city with that name, Nahor, in the distant past. Archaeology also states that that city was within the geographical area that is known as Mesopotamia. (ajaychhatria; 16 Apr 2009 05:39:36 on this thread. You may have to ask him for details.)
When Abrham asked his servant to go to his kindred, that only tells about the sons of Terah and their successors. There is another verse to make your point stronger, while agreeing with the other view. I have given enough hints on this.