As stated in Wikipedia,
Many electromagnetic coils have a magnetic core, a piece of
ferromagnetic material like iron in the center to increase the
magnetic field.The current through the coil magnetizes the iron, and
the field of the magnetized material adds to the field produced by the
wire. This is called a ferromagnetic-core or iron-core coil. A
ferromagnetic core can increase the magnetic field and inductance of a
coil by hundreds or thousands of times over what it would be without
the core. A ferrite core coil is a variety of coil with a core made of
ferrite, a ferrimagnetic ceramic compound.
Choice of core material thus affects the magnitude of the signal created for a given string vibration amplitude (not to mention some dependence on frequency). Then there's the "saturation" effect, where some materials reach their max possible magnetic field strength.
All this is to say that pretty much any metal that will sustain the magnetic field caused by interaction between the string's motion and the magnet (full core or button on top) and thus generate the desired AC electrical signal in the coil. Sound quality and gain and the like depend on how serious you want to get into the theory and practice of E&M .
If you go with some hardware-store screws, definitely pick ones which a magnet can latch onto. There might be some stainless-types out there which could be problematic.
There's a decent introduction to coil & core behavior, with pictures, at electronics-tutorials.