The first floppy disks were eight inches. This size was set by IBM; I haven't been able to find any indication of why they chose it, but maybe it was just because it seemed quite small to them compared to the hard disks they were used to.
In a 1976 meeting, An Wang of Wang Laboratories informed Jim Adkisson and Don Massaro of Shugart Associates that the 8-inch format was simply too large and expensive for the desktop word processing machines he was developing at the time. He argued for a $100 drive.
And that was how 5.25″ came to be.
I can see how the smaller size was considered preferable if e.g. you were hoping to build a portable computer. But why was it cheaper? Is it because of the engineering difficulty of making a mechanism both large and precise?