"Weak bits" are a means of copy protection that generates areas on a disk that read back as random values, without the floppy disk controller actually detecting an error. When copying such a weak bit to a new disk with a standard FDC controller, it will end up as a distinct 0 or 1 depending on the random state that it was read in and never change as long as the disk remains intact. The copy protection check will repeatedly read such locations and check whether the bit value actually changes on repeated reads. Obviously, that check needs to be well hidden in the software, otherwise, it could easily be deactivated by malicious people.
The apparently common conception that this would be done with "weak magnetization" or a change in magnetic flux that is "smaller" than normal, is wrong. While this might theoretically be possible (aging disks that can be read successfully after several re-tries seem to prove this), I very much doubt this could be reliably reproduced in mass-production of disks.
You can simply generate such weak areas by not formatting a specific track (or part of that track) on the disk at all. This will create a complete track of "weak bits" that are entirely unreadable by a normal FDC - Unfortunately, a software pirate can reproduce the same thing with a standard floppy disk controller by simply not formatting that track on his copied disk as well - So, this is maybe too simple.
What's instead done is described in detail in this article that describes in-depth the analysis of a copy-protected game disk for the Atari ST (Dungeon Master) - Scroll down about 2/3 of the page to where the action starts.
In a nutshell, the PLL on a disk drive controller relies on somewhat evenly spaced sync bits on the disk to sync its read window (that is, where it expects to be able to read a valid data bit) to the specific RPM of the drive. By deliberately placing some out-of-sync clock bits on the disk, placing a magnetic flux transition exactly where the FDC would assume a stable state and determine a data bit value it can be confused so that it will in fact "read" a random value (within normal RPM variations of the drive it will either "see" a flux transition there or not). A standard floppy disk controller will not be able to write such a timing violation to the disk, thus not be able to copy the "weak" bit, but rather copy it as a "normal", non-changing bit. You need a disk controller that can record a raw MFM data stream, complete with its timing. So, weak bits are actually not "weak magnetization", but a violation of the MFM timing.
Standard disk copying machines will not use a standard floppy disk controller and simply copy the low-level MFM data stream with the standard-violating sync bits as-is. The master (at least the affected track) must obviously be written with a specific disk controller that can write such violations. Modern equipment like Kryoflux can do that, apparently. I would expect that back in the days similar equipment existed.