Why does an electron have spin? An electron is assumed to be a point particle that does not have structure and volume. Why does it have spin? Does this imply that the electron has volume? It is hard to imagine that a point (without volume) can spin.
 A: When experiments started measuring the microcosm of particles it became necessary to invent quantum mechanics in order to build descriptive and predictive mathematical models. The model we have now is called the standard model . The elementary particles are postulated as point particles and the model has been validated innumerable times, by all the available data, including for the existence of spin and magnetic moment for the point particles in the table.
Basic "axiomatic" conservation laws ( because of the mathematical format)for this quantum mechanical model, are energy conservation, momentum conservation and angular momentum conservation. In addition special relativity  with its four vectors is basic in the standard model. 
The reason the particles in the table are assigned a spin is because of angular momentum conservation in particle interactions. If there were only orbital angular momentum and no intrinsic angular momentum for the particle the angular momentum would not be conserved. With the assignment of a specific spin for each particle the conservation of angular momentum holds ( is validated) for all existing data.
What is spinning in a point particle with no volume? It is the same as asking :"where is the charge on a zero volume particle?".
The exact answer is : nothing is spinning, and the charge is concentrated on a point. Mathematically everything is consistent by assigning spin and charge to an electron or any of the particles in the table.
It does not sit well with classical intuition, but most everything in quantum mechanics does not sit well with classical intuition, it is  a different framework, based on experimental numbers and a sophisticated mathematical model.
Intrinsic spin is a quantum number necessary for conservation of angular momentum. Intrinsic charge is a quantum number necessary for conservation of charge, and the particles are labeled by the values.
A: Here, I would like to present a rigorous point of view for the concept of spin. The phenomenological motivations are already given in anna v's answer and akhmeteli's answer.

First of all, spin isn't about structures, or volumes. It is about statistics (behaviors) of a field/particle, regardless of whether it has structure/volume or not.
Spin is not only unique to fermions, photon has spin, as well, just in terms of polarization. Just like polarization states of electromagnetic field, a fermion has up or down spin states. Both of them are intrinsic properties, just different statistics (behavior).
By saying spin is an intrinsic property, we mean that, considering the electron field for example, for each coordinate on spacetime there is another topological space that describes the spin behavior. Just as we describe spacetime in 4 real dimensions, this pointwise-attached spaces have 4 extra complex dimensions, each one is called fiber and the whole thing (the theory) is called fiber bundle (associated with a symmetry group - it is $SU(2)_L \times SU(2)_R$ in this case, where these symmetries are related to conservation of angular momentum). It's like each field at each position, besides the kinematic interactions, also has an inner world that describes their intrinsic behaviors.
So, instead of thinking there is a spacetime manifold and just particle fields at each point, we think there is a spacetime fiber bundle where each fiber is a spin space with its own coordinates. Fermions have components living in those coordinates, and that's why they are denoted as $\Psi_a (x)$, where $a=1,2,\dot1,\dot2$ are spinor index for left-handed particle, left-handed anti-particle, right-handed particle, and right-handed anti-particle, respectively. Another example, Dirac gamma matrices, $\gamma^\mu_{ab}$, which are the representation of fermion spin, have both spacetime and spinor components.
Actually we are already familiar with this kind of thinking when we first encounter electric and magnetic fields. 
At each point, there is an electromagnetic field, $\mathbf{E}(x)$ and $\mathbf{B}(x)$ (or $F_{\mu\nu}$ in the covariant formulation), where they have different set of values for each direction. They are called the curvature of the fiber space. So, every point on the spacetime has another space that even has a curvature, and what we actually see as an electromagnetic field is the interaction of the curvature of these fiber spaces to the charged particles, which are source of the curvature.
A: 
"An electron is assumed to be a point particle that does not have
  structure and volume."

This popular statement should not be accepted uncritically.
If you look at the papers offering strong experimental limitations on the size of electron (such as a review by the Nobel Prize winner H. Dehmelt in Physica Scripta. Vol. T22, 102-110, 1988), you will see that "point electron" there actually means "an electron that is described by the Dirac equation and QED with extremely high accuracy" (this is not a quote, this is just my summary; the actual quote confirming my summary is "an elementary Dirac particle, such as the electron, is the closest laboratory approximation of a point particle." See also Fig. 8 in the quoted article).
Thus, now we have a different question: does the Dirac equation describe a point particle? This is not obvious, as the Dirac equation is an equation of quantum theory, therefore, the uncertainty principle is valid for this equation. The Dirac electron can have a well-defined coordinate, but only at the expense of a poorly defined momentum. Therefore, you cannot say that an electron is "a point (without volume)" when we speak about spin (angular momentum). The uncertainty of angular momentum is of the order of the Planck constant, so there is no contradiction with the existence of spin.
Another thing. How do you define the size of the electron? The electron carries the Coulomb field. Is this electric field a part of the electron or not? (Cf. the notion of "dressed" electrons in QED). Remember that the radius of the Coulomb field is infinite. 
A: 
An electron is assumed to be a point particle that does not have structure and volume.

Perhaps the existence of the electrons electric charge could be explained by a pointlike electron. But for the electrons magnetic dipole moment the electron has to have some volume or extension. So nothing prevents me to imagine the electron as a gaseous volume. Rutherfords imagination about point like electrons could be obsolet after 100 years.

Why does it have spin? Does this imply that the electron has volume? It is hard to imagine that a point (without volume) can spin.

The electrons magnetic dipole gets aligned under motion and an external magnetic field in a way that it gets deflected (Lorentz force). It is a cyclical process of alignment, of emission of photons with deflection and disalignmt again, alignment, emission ... and so on until the kinetic energy gets exhausted (in emitted photons) and the electron stands still. 
A: Pointlikeness of electron is an inclusive picture. It is as pointlike as the Moon: when we content ourselves with a very simplified "image" - a point.
An electron is coupled to the electromagnetic field and it is felt far far away, as if it were extended. In QED it is indeed extendend in an elastic picture (see paper by T. Welton and by me).
On the electron spin, there is a paper of Ohanian (google it). It shows what is rotating in the QM description of electron.
