The spin of an electron was classically thought of a spinning ball of charge. We know that that is not the case in the quantum picture, as the electron is pointlike.
So why, then, do we still describe quantum spin as "intrinsic angular momentum?" Why isn't it just "intrinsic magnetic moment," or something else?
There are two main reasons that we describe spin as an 'intrinsic angular momentum':
Because it is an angular momentum. This comes primarily from a fundamental level, in that angular momentum is always the generator for rotations and the Noether charge that is guaranteed to be conserved if the theory is independent of orientation, and more generally angular momentum is always canonically conjugate to orientation. In all of those aspects, when it comes to electrons, the role of angular momentum is played by spin.
Now, those sound like a lot of heavy-handed terms, but by them I mostly mean: the role of angular momentum in classical as well as quantum physics goes well beyond describing spinning balls of stuff, and the identification of spin as an angular momentum within that framework isn't much affected by the loss of one minor component of the description.
That said, though, there's plenty of experimental evidence that spin really is convertible into the regular mechanical angular momentum of spinning balls of stuff, from the Einstein-de Haas effect onwards. If you don't include spin into your system's total angular momentum, then your angular momentum books will be unbalanced.
Because it is intrinsic. Generally, if a system has linear momentum $\mathbf P$, we distinguish between
The angular momentum of the Earth's orbital motion around the sun is of the former type, and the angular momentum of its rotation about its axis is of the latter; electrons' spin is of the second type.
Put those two components together, and the name "intrinsic angular momentum" is perfectly justified.
"angular momentum" is accepted because of the similarity with orbital angular momentum as explained below, and "intrinsic" means that we do not know what that "spin" actually is and where it comes from. And indeed, that's just a name, not meaning that there is really a rotation in the meaning of classical mechanics.
At the beginning, from the analogy that rotating charge generates a magnetic moment, people called that "spin angular momentum" and even imagined a real spin or rotation. That name has been accepted although they were wrong.
The reason for accepting the name, "intrinsic angular momentum", can be clarified in two side. On the one hand, spin has many same properties with orbital angular momentum, such as the commutation relation- the fundamental of quantum mechanics- between 3 spatial components, so it's "angular momentum". On the other, it's impossible for point particles to rotate in the meaning of Newton mechanics for many reasons and no experiment has discovered that rotation, so it's "intrinsic", like that the charge of a electron is "intrinsic", which often means that we do not know why it exists, where it comes from, and even what it is.
P.S.
Actually, the spin can be deduced from Dirac's QM, but that's another thing. Anyway, that name, "intrinsic angular momentum", can be understood from the point view of history.
The name is simply a question of technical English / Language usage. The name's motivation is foremostly that, even in the absence of an intuitive, everyday visual perception of "rotation", there is still a quantity that is conserved by dent of Noether's Theorem given the invariance of a Lagrangian formulation of mechanics under transformation of space by the rotation group $\mathrm{SO}(3)$, and latter's image under various representations (e.g. corresponding transformation groups on quantum state spaces).
So, from the abstract standpoint of Noether's theorem, the root of the phenomenon is still exactly the same as that of more "everyday" angular momentum, such as of a skier or an acrobat that just happens to be accompanied by a certain visual experience, so why call it something else?
Summarize this answer by asking yourself, "What if we had evolved as unsighted but clever beings? Should we still have a notion of angular momentum if we couldn't see?". Indeed, through Noether's theorem, we most assuredly should, although it might not have an "everyday" analogy as it does for sighted creatures.
The electron has a non-zero amount of mechanical angular momentum. This is demonstrated in a setup that uses the Einstein-de Haas effect. (I will abbreviate that to 'E-dH effect'.)
In an answer to a Stackexchange question about the E-dh effect contributor Gary Godfrey has described the usual setup as follows: "The spins of all the electrons in the cylinder are aligned by the magnetic field from the coil. Then the field is reversed so the electron spins line up the other way. This imparts angular momentum to the cylinder for each electron flipped. The effect on the cylinder is very small, so the flipping is repeated many times at the torsional resonant frequency of the cylinder on the fiber. This pumps the resonance up to some maximum deflection. Using the fiber spring constant, the fiber damping coefficient, and the moment of inertia of the cylinder you can calculate how much angular momentum per flip is being transferred to the cylinder."
(There is a 17 second youtube video uploaded by University of Osnabrück physics department, showing their E-dH effect setup in swing. )
About electron size:
With experiments that bring out the particle-like behavior of electrons one can arrive at an upper bound for the size for the electron-as-a-particle. As you are referring to in your question, that upper bound was found to be smaller than the minimum size that would be necessary to explain the magnetic moment being generated by spinning in the classical sense. That doesn't necessarily mean the electron is point-like. It just says: too small to be compatible with classical explanation.
One reason is by analogy - the algebra describing the spin of the particle coincides with that describing angular momentum (which a point like particle can also have independent of its spin)
The assignment of spins to particles preserves conservation of momentum,it makes the mathematical theories consistent with observations.
For example , the assignment of spin 1 to photons in e+e- --> gamma gamma:
The Born cross-section for the process e+e- -> gamma gamma (gamma) was determined, confirming the validity of QED at the highest energies ever attained in electron-positron collision
The calculations would not fit the data with 0 intrinsic angular momentum (spin) contributions from the photons.
I am not sure "spin is not actually rotation", and when we say that "the electron is pointlike", it probably should not be taken literally. Both statements ("no rotation" and "pointlike") are problematic because of the uncertainty principle.
