Yes, a HDD magnet is fine or any neodymium one and will destroy the data by sweeping such a powerful magnet across the surface of a disk. However, the reliability of such a procedure may vary.
You can also build your eraser in a similar manner to making a head demagnetizer for tapes (which is pretty simple: insulated wire, nail and an input voltage source).
If you erase your floppy disk (completely or partially) using a magnet, the track sector markers are destroyed. A quick format won't find them and therefore the format and/or any write to the disk will fail. A normal format may also fail due to not finding 1st sector marker information. However, special MS-DOS era programs may be able to re-format the disk and rebuild all track sector markers, therefore making the disk reusable.