Many (compact orientable) hyperbolic 3-manifolds have non-trivial $SU(2)$ representations.

By Mostow rigidity, the representation of the fundamental group $\Gamma$ of a closed hyperbolic 3-manifold into $SL(2,\mathbb{C})$ (lifted from $PSL(2,\mathbb{C})$) may be conjugated so that it lies in $SL(2,K)$, for $K$ a number field (because transcendental extensions have infinitesimal deformations in $\mathbb{C}$). In particular, the traces of elements will always lie in a number field. One may take different Galois embeddings of $K$ into $\mathbb{C}$, and get new representations of $\Gamma$ into $SL(2,\mathbb{C})$. Sometimes, this representation is just conjugate to the original (e.g. if $K$ was chosen too large), but in other cases the new representation of $\Gamma$ lies in $SL(2,\mathbb{R})$ or in $SU(2)$. A nice class of examples of this type are arithmetic hyperbolic 3-manifolds. In fact, they are characterized by the fact that all traces of elements are algebraic integers, and non-trivial Galois embeddings lie in $SU(2)$ (you have to be a bit careful about what this means). Some arithmetic manifolds will only have the complex conjugate representation this way (basically, if squares of the traces lie in a quadratic imaginary number field), but otherwise you get a non-trivial $SU(2)$ representation. The simplest example is the Weeks manifold, with trace field a cubic field. I suggest the book by MacLachlan and Reid as an introduction to arithmetic 3-manifolds. The description I've given though is encoded in terms of quaternion algebras and other algebraic machinery. Another characterization of arithmeticity is in this paper. The nice thing about these representations is that they are faithful.
There is a very explicit way to see these representations for hyperbolic reflection groups (studied by Vinberg in the arithmetic case). Basically, given a hyperbolic polyhedron with acute angles of the form $\pi/q$, sometimes one can form a spherical polyhedron with corresponding angles which are $p\pi/q$, and get a representation into $O(4)$. Passing to finite index manifold subgroups, one can obtain $SU(2)$ reps. (since $SO(4)$ is essentially $SU(2)\times SU(2)$).

There are other ways one has $SU(2)$ representations, but they are less explicit.
Kronheimer and Mrowka have shown that any non-trivial integral surgery on a knot has a non-abelian $SU(2)$ representation. Also, any hyperbolic 3-manifold with first betti number positive or a smooth taut orientable foliation has non-abelian $SU(2)$ representations.

**Addendum:** Another observation relating $SU(2)$ representations to hyperbolic geometry is via the observation that the binary icosahedral group (a $\mathbb{Z}/2$ extension of $A_5$) is a subgroup of $SU(2)$. By an observation of Long and Reid, every hyperbolic 3-manifold group has infinitely many quotients of the form $PSL(2,p)$, $p$ prime. These groups always contain subgroups isomomorphic to $A_5 < SO(3)$, so one may find a finite-sheeted cover which has a non-abelian $SO(3)$ and therefore $SU(2)$ representation. I have no idea though whether these representations are detected by the Casson invariant or Floer homology.