How to select the principal investigator in a group? If a group of researchers collaborating on a grant proposal, how the principal investigator can be selected? The researcher with the idea, the one who invited other researchers, the most well-known researcher in the group, the researcher with highest academic rank (e.g., full professor), etc? Should the group decide? What if the first researcher would like to be the PI?
I'm not sure why you are asking. But if it is because there are disputes within the group it might be best, overall, to find a different group. Clashes at that level don't portend success, but rather a lot of frustration. 

But all of your suggestions are valid (as is that of user Massimo Ortolano). If it is obvious that one of you would give the work an advantage for publication, do that. If one of your junior members needs a boost in reputation, do that (sorry Massimo). Anything can work as long as there is buy in from the group. If someone has a strong inclination to do it and no one else objects, he or she would be a good choice as they will likely commit to getting things done and organized. If someone has a history of successful grant writing, another good choice, unless the granting institution has rules limiting repeat grants. 

But the most important quality is agreement.