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 $\sigma/\sqrt{n}.$
"Monte Carlo convergence" means that you've sampled enough individuals to represent (and understand) a general population. If the probability models behind your Monte Carlo simulation are accurate, then your results will match reality as you increase your sampling size.

Monte Carlo convergence becomes difficult when you try to study a low-probability sub-population. As you sample the general population and only count those outcome that match your low-probability tally, you find convergence to be slow. Very few individuals arrive to your desire outcome. So, you sample and sample, but converge never comes.