Question: I have read about convergence in terms of MC simulation for derivative pricing, but I am not clear on what it exactly means.
Let us suppose I price an option 100,000 paths twice and both result in the same option price. Does that mean 100,000 paths has resulted in convergence?

Also, in determining the number of paths to use for pricing, is getting the same option price with 2 different runs a factor? (Assumption is I am not reseeding the Random Number. So the sequence of Random Numbers between the two runs is different).

Answer: 
the output of an MC simulation depends on the random numbers used and if the distribution used is not too weird, after 10,000 runs you will get an answer that is distributed
$$
\mu + \frac{\sigma}{\sqrt{n}} Z,
$$
with $Z$ a standard normal. Here $n=10,000.$

With $\mu$ the quantity you want and $\sigma$ the standard deviation. 
So you won't get precisely the same answer as you will get different values of $Z$ but if they are close together it is a good sign. 

The usual approach is to estimate $\sigma$ as well as $\mu$ and then take your error magnitude to be $\sigma/\sqrt{n}.$
With Monte Carlo, which is nothing more than a numerical method to approximate a definite integral, convergence to K significant digits using N samples means that you will obtain the same K significant digits regardless of the random number sequence used for the N samples.

When you say that you obtained the same option price from two Monte Carlo runs using 100,000 samples, I am presuming that you are truncating or rounding your Monte Carlo result to cents, or possibly dollars. Using 100,000 samples for a Monte Carlo, you can have numerical method errors ranging from at least +/- 6% of the answer you get. So, if you truncate your MC answers to 10% or more precision (to deciles), then it is very possible that you could get the same answers every time you run your MC using 100,000 samples. If you truncated your MC answers to less than a 10% precision (to say percentiles), you got lucky. It will be just a fluke that you got the same answers twice in a row.