What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwise pure? I'll freely admit that I have a rather hard time keeping straight different notions of purity of etale sheaves, and I think part of the problem is the lack of counterexamples.
For example, it's a theorem that the stalks of intersection cohomology sheaves (with coefficients in $\overline{\mathbb{Q}}_\ell$, say) are pure.  But here, pure means "looks like the cohomology of a possibly singular projective variety" i.e., the weights are bounded above, but not below.  
This is perhaps not so surprising: if, say, one takes constant coefficients, then the IC is a summand of the pushforward of a resolution of singularities, and so the stalks are summands of the cohomology of projective varieties.  
But, if you do geometric representation theory, a funny thing happens.  In practice, one seems to always get "pointwise purity" i.e. the stalk has the weights that one expects from the cohomology of a smooth projective variety.  There are various geometric reasons for this (for example, if your resolution of singularity is symplectic, or if the fibers of your map have a paving/$\alpha$-partition by affines or other pure varieties), but it makes it hard for me to imagine anything else.  I think my understanding of algebraic geometry would be improved by knowing an example of an intersection cohomology sheaf on (say) an affine variety which isn't pointwise pure.  

What is a good example of such an intersection cohomology sheaf?

 A: This isn't really a bona fide answer. It's more a series of thoughts, which perhaps
you or someone else can complete.
Take a normal surface singularity $(X,x)$, whose resolution (which exists in any 
characteristic) consists of a cycle of smooth  curves (with at least one having nonzero genus). The point is that the dual graph should have some first homology. I suspect that this may be an example of what you're looking for. My impression is that the singularities that show up in  representation theory tend to be rational and so quite far from this. 
Since I'm a complex geometer, let me say things over $\mathbb{C}$, but I suspect it works
$\ell$-adically. (I'm trying to learn the $\ell$-adic stuff, but I haven't quite got there.)
Set $IC= j_{!*}\mathbb{Q}[2]$, where $j:X-x\to X$ is the inclusion. The stalk of $H^{-1}(IC_x)$ is the first cohomology of the link of the singularity, and this won't
be pure as a Hodge structure. The assumptions ensure that both $W_0$ and $W_1$ are nonzero.
A: Here is an example along the lines of Donu's suggestion:
We work over a (large) finite field $k$. Let $C \subset \mathbb{P}^2$ be an irreducible  plane curve of degree $d$ and geometric genus $g > 0$ having at least one node. For example, the curve given by the equation
$$y^2 = (x^2-1)(x-a_1)\cdots(x-a_{d-2})$$
 with $d>4$, all $a_i$ distinct and nonzero and $char(k) \neq 2$.
Blow up $d^2 + 1$ distinct smooth points on $C$ to get a smooth surface $X$ and let $D$ be the strict transform of $C$. We have $D\cdot D = -1$ since $C\cdot C = d^2$. By Artin's contractability criterion, we can contract $D$ to obtain a normal surface $Y$ with a singular point $p$. Now $H^1(D, \mathbb{Q}_{\ell})$ has weights $0$ and $1$, the $0$ coming from the node and the $1$ since the geometric genus is positive. By proper base change and the decomposition theorem, it follows that $IC_Y$ is not pointwise pure (at $p$).
