Name of fastening metal pieces holding the cardboard back to a picture frame?

Points

Thin metal tabs used to hold the mat, mount board and/or glazing inside of wood picture frames. Some points are stiff while others are flexible to allow access into the frame.

https://www.framedestination.com/media/wysiwyg/points_1.jpg

If you're asking about something like this:

then they are clips. This image is from a page advertising "Wood picture frame spring clips" from webpictureframes.com.

Other designs are available, including those which fit into a slot in the frame. This type isn't very adjustable for different thicknesses of cardboard but they are still clips (as are the hanging clips also shown here):

(Image from dataliteframes.co.uk)