How can an infinitesimally small object rotate? How is it possible for an object like a black hole or electron to spin since they are made up of no other components? If my understanding is correct, an electron is an elementary particle which means that it is just a point in space, and a black hole is a point in space with a very large "assigned" mass.
 A: 
If my understanding is correct, an electron is an elementary particle which means that it is just a point in space, ...

The electron spin is a special case of the general concept of angular momentum, which is a physical quantity generated by rotations. This is completely analogous to energy being generated by time translations and momentum by spatial translations. An electron really is spinning. Mathematically, the Pauli matrices used for describing electron spin in the nonrelativistic regime generate a perfectly valid representation of the rotation group $\mathrm{SO}(3)$, although this representation cannot be built from tensor products of vectors.
Although you can't think of electron's angular momentum as caused by the velocity vectors of its constituent parts, which is a difference between the electron and the spin of macroscopic objects like baseballs or the Earth, there's absolutely no reason to require this kind of decomposition must be possible. The modern notion of angular momentum is what it means for something to spin, and whether the object is pointlike or not is irrelevant.

... and a black hole is a point in space with a very large "assigned" mass.

The singularity of a rotating black hole does is not pointlike, as e.g., the Kerr metric describing an isolated uncharged spinning black hole has a ring singularity instead. But it's actually inappropriate to think of the singularity of Schwarzschild black hole as a point in space: because the singularity not timelike, one can't think of it as a spatial point that endures in time. Because that singularity is spacelike, one could loosely say that it's a point in time instead.
A: An electron does not spin! Its intrinsic angular momentum (the so called spin), should not be confused with the point-like electron rotating in configuration space (then the gyromagnetic factor would be one which is in a way related to charge spinning in configuration space. Actually the gyromagnetic factor of the electron spin is approximately 2.)
A black hole is not point-like either (most attempts of a quantum theory of gravitation so far suggest there might not be a singularity in a black hole). And the total configuration of a black hole always includes the metric field (or space) surrounding the black hole, which obviously can carry angular momentum.
A: As the comment above says, the word "spin" should not be taken literally, as in the spin of a beachball.
The word spin came about as an attempt to physically understand the differing energy levels an electron can have, due to the magnetic field associated with it.
The idea behind it goes back to when the electron was discovered experimentally to have a different energy level than was expected theoretically.
It was never meant to be taken literally, merely as an analogy to explain what other source , in addition to orbital motion, could a particle have to explain this extra energy? The obvious choice was to imagine that the electron was spinning about an axis, at a very high speed, in much the same way as the total energy of the earth is not just it's orbital velocity, but it's rotational velocity as well. 
From Wikipedia:

The electron is a charged particle of charge (−1e), where e is the unit of elementary charge. Its angular momentum comes from two types of rotation: spin and orbital motion. From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field.

A: We don't know if the electron is an infinitely small object. It may have size and if it does maybe that will make you feel better about the fact that it creates a magnetic field due to its intrinsic spin. 
