In the padic case, is there any hope for a set of conditions on the local Langlands correspondence which would make it unique? In the case of GL(n) this is provided by L and epsilon factors. For classical groups, can one make a precise statement (even conjecturally), using lifting to GL(n)?
This answer will necessarily be full of conjecture, but I'll try to make things more concrete for some classical groups. For a reductive group $G$, and smooth irreducible representation $(\pi, V)$ of $G$, and Lgroup homomorphism $\sigma: {}^L G \rightarrow {}^L GL_n$, one may conjecturally associate a local Lfunction and $\epsilon$factor $L(\pi, \sigma, s)$ and $\epsilon(\pi, \sigma, s)$. As $\sigma$ varies over all Lgroup representations of ${}^L G$, this family of Lfunctions and $\epsilon$factors will uniquely determine the (conjectural) Langlands parameter $$\phi_\pi: W' \rightarrow {}^L G,$$ where $W'$ is the WeilDeligne group of the $p$adic field. Namely, there will be a unique $\phi_\pi$ such that $$L(\sigma \circ \phi_\pi, s) = L(\pi, \sigma, s), \epsilon(\sigma \circ \phi_\pi, s) = \epsilon(\pi,\sigma,s)$$ where the L and $\epsilon$ on the left sides are Artin Lfunctions (adapted to the WeilDeligne group instead of Galois group in a simple way). In this way, the L and $\epsilon$factors determine the Langlands parameter of a smooth irrep of $G$. This partitions the smooth irreps of $G$ into the "Lpackets". Now, for classical groups, one can get away with less information (or at least more computable information) in many cases. I refer to the recent paper of Wee Teck Gan and Takeda, "The Local Langlands Correspondence for Sp(4)" (avaialble on Wee Teck's home page, for example), where they prove that the Langlands parameterization is uniquely determined by L and $\epsilon$factors (and $\gamma$factors) of pairs (i.e. twists by $GL(n)$ with $n \leq 3$) for generic representations and a fact about Plancherel measure for certain nongeneric representations (this fact is analogous to a fact about Shahiditype Lfunctions, which are only understood in the generic case). For any classical group, I imagine one (probably not me) could come up with an analogous set of twists, etc.. to find a reasonable set of Lfunctions and $\epsilon$factors, and Plancherel measures to determine the Langlands parameterization. Perhaps this is done by others who prove functoriality for classical groups (P.S., Cogdell, Rallis, Jiang, Kim, Shahidi, and many others). Refining the local Langlands conjectures (I appreciate Vogan's survey most here), one should be able to parameterize the contents of an Lpacket (with parameter $\phi: W' \rightarrow {}^L G$) by the representations of a finite group $S_\phi$. This finite group has a general description, which in some nice cases (split, adjoint perhaps, I don't recall), is that $S_\phi$ is the component group of the centralizer of the image of $\phi$ in the dual group $\hat G$.) So to pin down the Langlands correspondence, one must characterize this parameterization of the elements of an Lpacket as well. For this, it seems that characters are key  there is certainly no telling the difference with Lfunctions or $\epsilon$factors. I believe this parameterization of the elements of an Lpacket should be uniquely determined by the stability of an associated sum of character distributions of elements of the Lpacket. I tend to avoid characters, but maybe the work of DeBacker and Reeder  who prove such a stability result for depth zero representations  is a good place to start. 

