This is a good explanation of what the numbers mean.
http://nerdblog.londondrugs.com/what-is-the-difference-in-all-the-wireless-n-speeds
N150s are capable of 150mbps. Way faster than what is needed for youtube. The salesperson is probably reading off of a guide that is set by the store. It's fine for videos. The major benefit to going to higher speeds is you are more likely to get higher quality adapters or adapters that can utilize multiple bands, giving you a better connection.
So it's not a bad idea to get an N300 or more. But it is not an absolute necessity for videos.
Edit: Just saw that you don't know what router she has. If she is broadcasting a Wireless G signal instead of an N signal, the network speed (not internet speed) is probably limited to 54Mbps. The difference between a N300 and N150 (at least the way I see it) would really only be the possibility of a higher quality adapter that I mentioned earlier.
If any of that is unclear let me know. I can try to explain it better