This is not an answer but a collection of observations, some of which, are, hopefully, interesting.
First, recall that Cheeger constant $h(X)$ for a complete open Riemannian manifold $X$ is
$$
\inf_D \frac{|\partial D|}{|D|}
$$
where infimum is taken overall bounded domains $D\subset X$ with smooth boundary and vertical bars denote volume (of appropriate dimension). In the context of infinite connected graphs, the same definition applies. Then $h(X)=0$ iff $X$ is amenable. Cheeger constant is also defined for closed manifolds (finite graphs) where one looks at the infimum of the quantity
$$
\frac{|S|}{|M_S|}
$$
where $S$ is a cooriented hypersurface in $M$ and $M_S$ the "positive side" of $S$ in $M$, and the infimum is taken over all cooriented hypersurfaces $S$. A sequence of closed connected manifolds $M_i$ (of "uniformly bounded geometry", i.e., constant dimension, with fixed bounds on curvature and a lower bound of the injectivity radius) is an "expander" if $h(M_i)$ is uniformly bounded below, away from zero, while diameters of $M_i$ diverge to infinity. Same definition applies to graphs where bounds on geometry are replaced by uniform bounds on valence. In other words, in the context of finite graphs or sequences of closed manifolds of bounded geometry, being an "expander" is essentially the negation of amenability. The question raised by Jon is if one can define an expander in the context of a single infinite graph or, I would say, a single open Riemannian manifold $X$ of bounded geometry. Defining an expander as before, by saying $X$ is nonamenable, is too weak. The usual link between expanders and open Riemannian manifolds/infinite graphs is via taking quotients: If $\Gamma_i$ be a infinite sequence of finite-index subgroups of a fixed discrete cocompact subgroup $\Gamma\subset Isom(X)$, then the sequence of quotients $M_i:=X/\Gamma_i$, is an expander. One then could say that $X$ itself an expander if it admit a sequence of group actions $\Gamma_i$ satisfying the above conditions. Note that the quotient manifolds $M_i$ "approximate" $X$, provided that intersection of the subgroups $\Gamma_i$ is trivial.
The question is how to eliminate groups in this definition, since some infinite f.g. groups with property T could be simple or at least, contain no proper finite-index subgroups.
I will now proceed in the category of Riemannian manifolds, but the discussion is equivalent to the one in the case of graphs. Let $Q_i\subset X$ be fundamental domains for the groups $\Gamma_i$. Let $S\subset M_i$ be a cooriented compact hypersurface, transversal to the projection of the boundary of $Q_i$, and let $T\subset Q=Q_i$ be preimage of $S$ in $Q$. Note that $T$ is not properly embedded in $Q$.
For each $T\subset Q$ we have the ratio
$$
\rho_{T,i}=\frac{|T|}{|Q_T|},
$$
where $Q_T$ is "positive side" of $T$ in $Q$. Then,
$$
\inf_{T,i} \rho_{T,i}
$$
is still bounded away from zero, provided that $\Gamma$ has property T. Here infimum is taken over all $T_i\subset Q_i$, so that $T_i$ is the preimage of a closed surface in $M_i$. (Yves exercise amounts to saying that $\Gamma$ having property T depends only on $X$ and not on the particular group acting cocompactly on $X$.)
One then can attempt to promote this to a definition: We say that a sequence of domains $Q_i\subset X$ is "anti-Folner" if the infimum of ratios $\frac{|T|}{|Q_{i,T}|}$ defined as above (using arbitrary cooriented hypersurfaces $T\subset Q_i$) is positive and so that $diam(Q_i)\to \infty$. One then would say that $X$ is an expander itself if it admits an anti-Folner sequence of connected bounded domains $Q_i\subset X$.
One can give a "spectral" interpretation of the anti-Folner property. Namely, for each $Q$ define
$$
\mu(Q):=\inf \frac{ \int_Q |\nabla f|^2}{\int_Q |f|}
$$
where infimum is taken over all smooth functions on $Q$. Then as in the proof of Cheeger's theorem, one gets: If $\inf \mu(Q_i)>0$ then the sequence $Q_i$ is anti-Folner.
Note that amenability of $X$ is equivalent to $\lambda(X)=\inf \lambda(D)=0$, where $\lambda(D)$ is the lowest positive eigenvalue of a bounded domain $D\subset X$ and
$$
\lambda(D)= \inf \frac{ \int_Q |\nabla f|^2}{\int_Q |f|},
$$
where infimum is taken over all smooth compactly supported functions on $D$. So the difference between $\lambda$ and $\mu$ is that in the definition of $\mu$ we allow functions in bounded domains in $X$ that do not have compact support.
What's missing here is a proof that $\mu$ is bounded away from zero in some interesting examples, since, in the context of sequences of discrete groups $\Gamma_i$, I was using not all but only some hypersurfaces in $Q_i$ (the ones which came from compact hypersurfaces in $M_i$).
