What is the interpretation of a non-diagonal density matrix? Suppose an ensemble of a spin $\frac{1}{2}$ particle in the $S_z$ basis is described by a diagonal density operator $$\rho=w_{11}|+\rangle\langle+|+w_{22}|-\rangle\langle-|$$ where $w_{ii}$ are real and non-negative. In this ensemble, measurements of $S_z$ will give the value $S_z=+\frac{1}{2}$ with a probability $w_{11}$ and the value $S_z=+\frac{1}{2}$ with a probability $w_{22}$, and $w_{22}=1-w_{11}$.
What is the interpretation of the coefficients $w_{ij}$ of a nondiagonal density matrix of the form $$\rho=w_{11}|+\rangle\langle+|+w_{22}|-\rangle\langle-|+w_{12}|+\rangle\langle-|+w^*_{12}|-\rangle\langle+|$$ where $w_{ij}$ for $i\neq j$ can in general be complex?
 A: The answer is implicit in the general principles, so I'll review the general principles first.
The observable for any component of the particle's spin can be written $\hat u\cdot\vec S$ where $\hat u$ is a unit vector and $\vec S=(S_x,S_y,S_z)$. For any unit vector $\hat u$, this observable has eigenvalues $\pm 1/2$, so we can always write this observable as
$$
 \hat u\cdot\vec S = \frac{1}{2}P + \left(-\frac{1}{2}\right)(1-P)
\tag{1}
$$
where $P$ is the projection operator onto this observable's spin $+1/2$ subspace  and $1-P$ is the projection operator onto its spin $-1/2$ subspace. When the observable $\hat u\cdot\vec S$ is measured, the probabilities of the possible outcomes are
\begin{align}
 \text{probability}\Big(\hat u\cdot\vec S \to +1/2\Big) 
  &= \text{trace}(P\rho )
\\
 \text{probability}\Big(\hat u\cdot\vec S \to -1/2\Big) 
  &= \text{trace}\big((1-P)\rho \big),
\tag{2}
\end{align}
where the left-hand sides use a notation that is hopefully clear. Equations (1)-(2) are basis-independent. 
The OP uses a basis that diagonalizes $S_z$. Equations (2) imply that if we measure $S_z$, then the probabilities of the outcomes $+1/2$ and $-1/2$ are $w_{11}$ and $w_{22}$, respectively. The same equations imply that if we measure $S_x$ or $S_y$, then the probabilities of both outcomes depend on all of the components of $\rho$, including the complex off-diagonal component $w_{12}$. 

What is the interpretation of the coefficients $w_{ij}$ of a nondiagonal density matrix of the form [shown in the OP] where $w_{ij}$ for $i\neq j$ can in general be complex?

One way to answer this is that the probabilities (2) generally depend on $w_{12}$. The only exception is when $\hat u=(1,0,0)$ so that $\hat u\cdot\vec S=S_z$. 
If a more specific "physical" interpretation of $w_{12}$ by itself is really desired, here's one example of such an interpretation:
\begin{align}
 2w_{12} 
 &=
 \text{probability}\Big(S_x \to +1/2\Big) 
\\
 &-
 \text{probability}\Big(S_x \to -1/2\Big) 
\\
 &+ i\,
 \text{probability}\Big(S_y \to +1/2\Big)
\\
 &- i\,
 \text{probability}\Big(S_y \to -1/2\Big).
\tag{3}
\end{align}
To check this, use the fact that (in one convention) the matrices
$$
 \frac{1}{2}
 \left(\begin{matrix} 
  1 & 1 \\ 1 & 1
 \end{matrix}\right)
\hskip2cm
 \frac{1}{2}
 \left(\begin{matrix} 
  1 & i \\ -i & 1
 \end{matrix}\right)
\tag{4}
$$
project onto the $+1/2$ eigenspaces of $S_x$ and $S_y$, respectively, together with
$$
 \rho = 
 \left(\begin{matrix} 
  w_{11} & w_{12} \\ w^*_{12} & w_{22}
 \end{matrix}\right).
\tag{5}
$$
The interpretation (3) looks unnatural because attempting to interpret specific components in a specific basis is technically always unnatural. The important message is that any valid physical interpretation must follow from equation (2). The physical content of those equations is basis-independent, just like the physical content of general relativity is coordinate-independent.

A comment below the question asked about a relationship between eigenvalues and purity, so I'll address that, too. The state is pure if and only if $\rho^2=\rho$, which is equivalent to the condition that $\rho$ have one eigenvalue equal to $1$ and another equal to $0$ (because the trace of $\rho$ must be $1$). The eigenvalues of $\rho$ depend on all of its components, not just on the diagonal ones, so the purity/impurity of the state also depends on all of its components.
