What sorts of mathematical statements are predicted by the AdS/CFT correspondence?
My "understanding" (term used very loosely) is that this correspondence isn't a mathematically rigorous statement, but I can still imagine that certain baby cases might spit out interesting well-defined identities or relationships. I'm very happy for this question to be broadly interpreted.
Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of this correspondence.
An example of nontrivial mathematical prediction, using a baby version of AdS/CFT (the Caffarelli-Silvestre extension) is the conformal invariance of the scaling limit of critical long-range Ising (or $\phi^4$) models in 3D. You can find this discussed in the article "Conformal Invariance in the Long-Range Ising Model" by Paulos, Rychkov, van Rees and Zan. For a precise formulation of the prediction as a mathematical conjecture see, e.g., my article "Towards Three-Dimensional Conformal Probability", journal version, preprint version.
Also note that a lot of work has been done as far as making CFT in 2D rigorous, most notably for the Ising model. Carlo mentioned the Ryu-Takayanagi formula. This is perhaps something one could prove (maybe someone did that already, I don't know).
Good to know also: A mathematical result which has the flavor of a very baby AdS/CFT correspondence is the representation of a temperate distribution in $\mathscr{S}'(\mathbb{R}^d)$, using the wavelet transform, as an integral over the Euclidean AdS, i.e., the hyperbolic space $\mathbb{H}^{d+1}$, with the right invariant metric/measure. See for example the book "Wavelets, an Analysis Tool" by Holschneider. If one introduces a cut-off hypersuface approaching the conformal boundary in order to restrict the domain of integration in $\mathbb{H}^{d+1}$, one can use this operation as a way of regulating probability measures on $\mathscr{S}'(\mathbb{R}^d)$ (e.g. the Euclidean functional integrals of a CFT on the boundary). One can show these cut-off measures converge weakly to the given probability measure, as the cut-off surface approaches the boundary.
Since the AdS/CFT correspondence links quantum field theory to something as exotic as quantum gravity, I don't think there is any hope for precise mathematical statements coming out of that correspondence in the foreseeable future.
If I interpret the question as "explicit computational consequences of the AdS/CFT correspondence", then perhaps the Ryu–Takayanagi formula stands out. This formula for the entanglement entropy in a quantum field theory can be derived from the AdS/CFT correspondence, and checked with explicit calculations in some cases.
From the physics perspective, the strongest observational consequence of the AdS/CFT correspondence is the Kovtun–Son–Starinets bound on the viscosity/entropy ratio, which can be measured in a quark-gluon plasma.
I am not an expert in this area but I know that Costello, Gaiotto, Paquette, and others have been studying topological and holomorphic twists of ADS/CFT. Unlike the full correspondence this one seems amenable to mathematical analysis.
New exact solutions in classical general relativity have been found using AdS/CFT methods. It may sound trivial since there are already encyclopedias of exact solutions, but those new ones have quite unexpected properties. Some references to those works are found e.g. in sec. 4.2 of Hubeny's notes (in the same document you encounter an extensive catalog of applications of AdS/CFT to problems in mathematical physics).
