A possible generalization of the homotopy groups.
There's always information to be got. But in this case:
Based homotopy classes of maps $T^2\to X$ don't form a group! To define a natural function $\mu\colon [T,X]_*\times [T,X]_*\to [T,X]_*$, you need a map $c\colon T\to T\vee T$ (where $\vee$ is one point union). And if you want $\mu$ to be unital, associative, etc., you'll want $c$ to be counital, coassociative, etc. For $T=T^n$ with $n\geq2$, there is no $c$ that is counital. (The usual way to see this is to think about the cohomology $H^*T$ with its cup-product structure.)
The inclusion $S^1\vee S^1\to T^2$ gives a map $$r\colon [T^2,X]_* \to [S^1\vee S^1,X]_*\approx \pi_1X\times \pi_1X.$$ The image of this map will be pairs $(a,b)$ of elements in $\pi_1X$ which commute: $ab=ba$. It won't usually be injective; so there might be something interesting to think about the in preimages $r^{-1}(a,b)$.
Back in the 1940's, Ralph Fox defined something called the torus homotopy group. For a based space $(Y,y_0)$ and natural number $r$, the $r$-dimensional torus homotopy group $\tau_r(Y,y_0)$ is just the fundamental group of the mapping space ${\rm map}(T^{r-1},Y)$, based at the constant map (where $T^{r-1}$ is of course a torus).
The group $\tau_r(Y,y_0)$ contains isomorphic copies of $\pi_n(Y,y_0)$ for all $n\leq r$. Also, Whitehead products become commutators in the torus homotopy group. By passing to the limit over $r$ one obtains the (infinite) torus homotopy group $\tau(Y, y_0)$, which contains all of the homotopy information of $Y$ in one place!
Unfortunately for Fox, the idea doesn't seem to have caught on (although I hear he had a few others which did). MathSciNet only turns up 11 papers containing the phrase "torus homotopy groups" (although the most recent is from 2007).
Your problem is that $T^n$ is not in general a co-Moore space. Therefore Eckmann-Hilton duality breaks down, as the dual spaces no longer form a spectrum, and there would be no (co)homology theory dual to such a "homotopy theory". Thus, a theory of homotopy classes of pointed maps from $T^n$ to $X$ would be much less interesting than a theory of homotopy classes of pointed maps from $S^n$ to $X$.
On the other hand, the study of homotopy classes of pointed maps from a co-Moore space other than $S^n$ to $X$ does lead to useful theories of homotopy with coefficients. I believe these classify $X$ up to homotopy equivalence.