What’s so floppy about floppy disks?

Solution 1:

There are 3 origins to choose from. None of the references look very solid.

The 5.25-inch disks were dubbed "floppy" because the diskette packaging was a very flexible plastic envelope, unlike the rigid case used to hold today's 3.5-inch diskettes.

References:
http://www.answers.com/Q/Why_is_a_floppy_disk_called_floppy
http://computer.howstuffworks.com/floppy-disk-drive1.htm

floppy disk (named so because they were flexible)

Reference from obsolete website royal.pingdom.com

Origin of floppy disk: in contrast to a hard disk, which is rigid

Reference:
http://www.yourdictionary.com/floppy-disk

In the end this article from IBM suggests the term originates from the flexibility of the medium, but it is never stated outright.

The team considered using magnetic tape first, but then, in a project code-named “Minnow,” they switched to using a flexible Mylar disk coated with magnetic material that could be inserted through a slot into a disk drive mechanism and spun on a spindle. “I had no idea how important it would become and how widespread,” recalls Warren L. Dalziel, the lead inventor of the floppy disk drive. The first floppies were 8-inch disks that were bare, but they got dirty easily, so the team packaged them in slim but durable envelopes equipped with an innovative dust-wiping element, making it possible to handle and store them easily.

Reference:
http://www-03.ibm.com/ibm/history/ibm100/us/en/icons/floppy/

Solution 2:

Magnetic disks were traditionally made of cast aluminum, and were very hard and rigid (the same as the "hard drives" that one can still buy today, but much much larger). Think of circular saw blades without teeth.

In contrast, floppy disks were made of very thin plastic, which if held at the edge would bend under their own weight.

In use, they were kept flat by the force generated by their spinning.

For anyone familiar with standard hard drive disks, taking one out of its protective case revealed something so different from the precision machined aluminum platters, that they could hardly call them anything but "floppy". (Well, "flimsy" might have worked too, but that would also have implied unreliability.)