Why might it be important to Magwitch that Pip “always bear the name of Pip”?

In my opinion, this is because Magwitch wants Pip to change, but he does not want Pip to change in the most important ways.

Obviously, Magwitch wants to remake Pip's life. He wants to give Pip opportunities -- to give him a chance at a better life. But at the same time, he thinks that Pip is a good person and he does not want him to change. He wants Pip to remain the same person that he is at the point when they meet. I think that he wants Pip to keep that name so that Pip does not forget who he really is.

We can see that Pip has problems doing this over the course of the book, but in the end he becomes pretty much the same good person he was at first.