Best approach to suggest a PhD student from a external collaborator to be her/his PhD co-advisor
I worry about the politics of it and urge a bit of caution. The relationship between the student and the formal advisor is essential to completion of the degree. The politics arises because there is the issue of the personality of the formal advisor and how s/he would interpret such a "move". I think it would be vastly different for different people. Some would take it as very helpful. Others as interference. 

I think the best way is to propose it jointly to the student and the advisor and using somewhat tentative language. "I'm happy to help out there. Is there some way that we can come to a formal arrangement?" 

You don't want the student to be conflicted. You don't what the advisor to be annoyed. And most of all, you don't want the student to be getting different, possibly conflicting, advice from you and the formal advisor. 

However, there isn't really anything wrong with your intent. Just be sure you know the personalities and what the worst case scenario might be. Eyes open.