What is the evidence that the bones found under St. Peter's Basilica are actually St. Peter's bones? There were some bone fragments found under St. Peter's Basilica in the 1940's. Pope Paul VI said in 1968 that the bones were "identified in a way that we can consider convincing." What is the convincing argument that these bones are indeed's St. Peter's?

A perfect answer will include the details about how the bones were found.



On a related note, the bones have recently been made available for public viewing for the first time. During a Mass, Pope Francis seemed to venerate the relics, which may mean that the Church will officially declare them as St. Peter's bones.
A lot of the evidence is circumstantial, so it's probably a matter of opinion whether it's convincing or not:


The bones that are thought to be St. Peter's were found near a grave under the current high alter of the Basilica - a location that's long been considered St. Peter's grave.
There are written records that indicate that ~160 AD there was a shrine erected in the Vatican area over Peter's grave.
Excavations below the current high altar revealed the remains of the ~160 AD shrine and some evidence of an even earlier one (~70 AD).
Constantine was certainly convinced of the location of the grave when he built the first St. Peter's.  In order to position the centre of the apse in the right place, a large area of the Vatican hill had to be flattened. 
It's been quite a while since I've read any of the books about this... the bones thought to be St. Peter's weren't found in the grave, they were found in a niche in a wall next to the grave, a wall that has ancient graffiti carved into it that says something like "Peter is here."
The bones that were found in the grave were identified as being from an elderly man, heavily built.  There was dirt on the bones that matched the dirt in the grave and the theory is that the bones were moved to protect them since drainage and flooding of the lower levels of the Basilica was a concern.


Unfortunately, I think the details that would make this closer to the "perfect answer" you asked for run afoul of one of the StackExchange guidelines for what not to ask:  


  "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much."


Among the books that cover this are:

http://www.amazon.com/Bones-Peter-John-Evangelist-Walsh/dp/1933184752
http://www.goodreads.com/book/show/2438024.The_Tomb_of_St_Peter
http://www.amazon.com/Saint-Peters-James-Lees-Milne/dp/1199553344

The texts of (and some pictures from) these 3 books can be found on this site.