Eigenstates of time dependent Hamiltonians I am trying to figure out how to make sense of a time dependent Hamiltonian. In the Schrödinger picture, the one dimensional Hamiltonian is written:
$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t).$$
If the Hamiltonian is time independent, then it makes sense that any $\hat{H}$ eigenstate is a stationary state (consider evolved in time state by the unitary time evolution operator). This is the case when solving the TISE. However, I am incredibly confused about how to properly talk about time dependent Hamiltonians.
How am I to make sense of the potential being time dependent. Hence, the Hamiltonian operator (which in the Schrödinger picture is supposed to be time independent) being time dependent?
Additionally, what do the eigenstates of this Hamiltonian look like? And, are they stationary states?
 A: If the time-dependent part of the Hamiltonian is small, Dirac's time-dependent perturbation theory is appropriate  for solving the TDSE.
You have correctly identified that the TISE trick used for time independent hamiltonians underlying the systematic solution of the TDSE fails.
To address the question about the eigenstates and eigenvalues of an (), why bother? In any case, solving the analog of the TISE eigenvalue equation still produces E(t) eigenvalues and ψ(t) eigenvectors, nothing will be stationary, and, crucially,  not that useful, as it does not lead effortlessly to TDSE solutions. For  "academic" completeness, here is an example.
The 1D nondimensionalized ($\hbar=1$, $m=1/2$) toy equation you might look at is the ground state of the time-dependent oscillator,
$$
(-\partial_x^2 +x^2\cos(\omega t))\psi(t)=E(t)\psi(t).
$$
A "ground state" eigenvector and eigenvalue are
$$
\psi(t)(a/\pi)^{1/4} e^{-a x^2/2}, \qquad E(t)=a(t)=\sqrt{\cos\omega t },
$$
but you may check that this $\psi(t)$ does not help you get a solution $e^{-iE(t)t}\psi(t)$ of the TDSE; won't work. You might as well start from the beginning with the TDSE...
A: In nature, if you include everything in your Hamiltonian, there will be no time dependence (in the Schrödinger picture, as you said). That's one-to-one with time-translation invariance.
The reason a Hamiltonian may be time-dependent even in the Schrödinger picture is only that we are modeling a changing system just through a potential rather than including it as additional coordinates $x_2, x_3, ...$. This may be easier to treat than solving the entire two- or multiple-body problem.
You're not new to doing this, as it is done in time-independent potentials too. For example the harmonic oscillator potential doesn't just exist on its own in nature, it is a model for larger systems whose DOF we are too lazy to handle. For example it could be taken as an approximate potential for a particle stuck in a lattice; we get results without having to actually model the many degrees of freedom ($x_2, x_3, ...$) of the other particles. If that lattice is stationary, there need not be time dependence in the potential. But consider the situation where the lattice spacing changes over time and its center moves around. Then we'd need a time-dependent potential to model it. Again strictly the most accurate model would be with a time-independent potential including all particles involved, but it's postulated that a time-dependent potential will give similar enough results.
The eigenstates of a time-dependent Hamiltonian will change in time and depend on the exact potential function.
A: 
How am I to make sense of the potential being time dependent. Hence, the Hamiltonian operator (which in the Schrödinger picture is supposed to be time independent) being time dependent?

Time-dependent Hamiltonians usually arise from the application of a time-dependent field, for example, if an atom interacts with the electric field of light, the electric field oscillates in time, resulting in a time-dependent interaction.

Additionally, what do the eigenstates of this Hamiltonian look like? And, are they stationary states?

The instantaneous eigen-states of the Hamiltonian do not, generally, provide a valid direction to solve the TDSE. With perhaps a few exceptions (e.g., Rabi oscillations), solutions can only be obtained under some approximations. For example, if the the Hamiltonian changes slowly enough, one can employ the adiabatic approximation, where the state does follow the instantaneous eigen-state of the Hamiltonian. Another useful case occurs when the time-dependent portion of the Hamiltonian is relatively small, so that time-dependent perturbation expansion can be used.
