Is there any way of converting an existing field to a Matrix?
Unless each time an Entry is selected the name needs to be different (to other times that same Entry is chosen), then I would suggest using your latter approach of adding an secondary Title field as it would be more efficient in the long term because the secondary Title would only need to be added once. 

If it's about an Entry, consider keeping it 'with' the Entry. 

I use a similar approach for Meta Titles to override the Entry title for SEO purposes.