salukigirl, what field are you in? In some fields, it is looked down upon if you do your PhD in the same place as your undergraduate degree. And that might hamper a person in their job search later for professorships, if they were to stay in Academia. But I have never heard the same said about doing Master's and PhD in the same place. (People in the field I got my PhD in don't do Master's degrees, so this possibility didn't come up.) It is probably very field-specific. How are things viewed in your field? You might want to discuss this with the professor you work for.

You say one downside would be that you wouldn't get to work with new people. If you do stay in place, you could expose yourself to other researchers by participating in the research report meetings, their journal clubs, etc. You could also to a collaboration with another professor's group, and in that way get exposed to another approach and more people. Also, if your field is one in which you need to do postdoctoral research, then you can go somewhere else for that.

If your BF has a good job where you live that he does not want to give up, then that lends more weight to the stay option. This would be especially true if he works in a field that it is hard to find jobs in.