Many examples can be exhibited using a theorem of Gersten:
Theorem (Gersten): Let $G$ be a hyperbolic group of cohomological dimension 2. Every finitely presented subgroup $H$ of $G$ is hyperbolic.
This is already very interesting, but the question asked for examples where every finitely generated subgroup is word-hyperbolic. Recall that $G$ is called coherent if every finitely generated subgroup is finitely presented.
Corollary: If $G$ is word-hyperbolic, coherent, and of cohomological dimension 2, then every finitely generated subgroup of $G$ is hyperbolic.
We can now search the literature for examples of such groups:
If $\Gamma$ is the mapping torus of a free-group endomorphism then it is 2-dimensional, and coherent by a theorem of Feighn--Handel. It is hyperbolic whenever the endomorphism is atoroidal, by work of Brinkmann (for automorphisms) and Mutanguha (in the general case). (Giles also mentions these in his answer.)
A one-relator group $\Gamma=F/\langle\langle w\rangle\rangle$ has (virtual) cohomological dimension at most 2 by Lyndon and Cockroft, so in the context of your question we need to understand which are hyperbolic and which are coherent. Baumslag conjectured that such $\Gamma$ is always coherent, while Gersten conectured that $\Gamma$ is hyperbolic unless it contains a Baumslag--Solitar subgroup. These conjectures have been confirmed in certain cases:
If $\Gamma$ has torsion (i.e. $w$ is a proper power) then $\Gamma$
is hyperbolic by the B.B. Newman Spelling theorem, and is coherent
by recent work of Wise and Louder--W.
If $\Gamma$ is 2-free, i.e. every 2-generated subgroup of $\Gamma$ is free, then $\Gamma$ is hyperbolic by very recent work of Linton, and coherent by another recent paper of Louder--W.
- It is well known that random groups
$\Gamma = \langle a_1,\ldots, a_m\mid w_1,\ldots,w_n\rangle$
are a.a.s. hyperbolic in many regimes. They are also coherent by a recent paper of Kielak--Kropholler--Wilkes whenver $n<m-1$, and coherent with positive probability when $n=m-1$. It is reasonable to conjecture that they are still a.a.s. coherent when $n=m-1$, but a.a.s incoherent when $n\geq m$, and also in the density model.
As far as I know, the membership problem remains open in all these cases. It certainly isn't known for general locally hyperbolic groups.