Why electrons have spin one half?

If I'm interpreting this question correctly, I think this is actually a great question which does, in fact, have a definite answer. Essentially it seems like you're looking at the properties of fundamental particles and wondering why they seem so arbitrary. Spin stands out, as compared to mass or charge, I think because spin is an exact number with no error margins associated to it. This gives it a completely different feel from things like charge and mass.

First, there's some background information which I think is needed to properly frame the answer to this question. First we need to understand that all the modern models of physics are based in quantum field theory (which is essentially if you take quantum mechanics, but all your operators are allowed to depend on both space and time, rather than just time), and specifically based on relativistic quantum field theory.

What this means is that we have some states in our Hilbert space and we also have some operators which implement Lorentz transformations (a detailed, but perhaps difficult to parse source on this would be chapter two of Weinberg's QFT book). The generators of translations in Minkowski space are the components of the 4-momentum operator, and it can be shown that all these components commute with each other. Hence we can find a basis of simultaneous eigenstates for all four of these operators, each state being labeled by the 4-momentum of that state.

It is possible though that the 4-momentum alone is not sufficient as a set of quantum numbers to specify a complete basis on the Hilbert space, so there may be some other label on the states meaning the states can all be written as $|p,sigmarangle$ where $p$ is the 4-momenta of the state and $sigma$ is a label corresponding to whatever other quantum numbers are needed.

The sort of miraculous thing is that it can be shown that for massive particles (a similar argument works for massless particles) that these $sigma$ labels must correspond to a representation of $SO(3)$, which is the group of rotations in 3 dimensions. I wrote an answer about this argument here.

Now, it can also be shown that the Lie algebra of $SO(3)$ and $SU(2)$ are identical (there are some global differences between the two groups which we won't worry about), and all the representations of $SU(2)$ can be completely classified by the value of the largest eigenvalue of $S_z$, the $z$-component of the spin operator (which generates rotations), and furthermore these eigenvalues are only allowed to take either integer or half-integer values.

So then to take stock, we have shown that as soon as we demand special relativity (which is Lorentz invariance), we are forced into including these representations which are classified by spin!

What this means in the end is that when we sit down to try and build a theory, question we need to ask ourselves is not why we should include a particle of spin-$1/2$, but "how many spin-$0$, how many spin-$1/2$, how many spin-$1$" and so on. From here we already know we should start looking for things with spin, so experiment only has to tell us "of spin-$1/2$ particles, we have seen n-number of distinct masses" so to write down the theory we can say ah, let's build something with that number of spin-$1/2$ particles.

In some respects this is a simplification of the real procedure and there are additional concerns that might come up. But at the heart of it all, this is severely restricts the possibilities we need to consider and makes large swaths of the theory-creation process fixed ahead of time.