This answer is largely a rendition of Sridhar's comment into lambda calculus. A strong monad $T$ has the following introduction and elimination rules in the lambda calculus. 

$$
\frac{\Gamma \vdash e : A}
     {\Gamma \vdash \mathrm{val}(e) : T(A)}
$$
$$
\frac{\Gamma \vdash e : T(A) \qquad \Gamma, x : A \vdash e' : T(B)}
     {\Gamma \vdash \mathrm{let\;val}(x) = e \;\mathrm{in}\; e' : T(B)}
$$

You may recognize these rules as typing a variant of the do-notation in Haskell. 

Strength is needed to interpret the elimination rule, since the context $\Gamma$ is available in both premises of the elimination rule. Taking $\sigma : \Gamma \times T(A) \to T(\Gamma \times A)$, we can calculate:

$$
\begin{array}{lcl}
e & : & \Gamma \to T(A) \\\
e' & : & \Gamma \times A \to T(B) \\\
T(e') & : & T(\Gamma \times A) \to T^2(B) \\\
T(e'); \mu & : & T(\Gamma \times A) \to T(B) \\\
\langle id; e\rangle & : & \Gamma \to \Gamma \times T(A)\\\ 
\langle id; e\rangle; \sigma & : & \Gamma \to T(\Gamma \times A)\\\
\langle id; e\rangle; \sigma; T(e'); \mu & : & \Gamma \to T(B)\\\
\end{array}
$$

Without the strength $\sigma$, we could not use the context $\Gamma$ in $e'$. That is, we would get introduction and elimination forms:

$$
\frac{\Gamma \vdash e : A}
     {\Gamma \vdash \mathrm{val}(e) : \Diamond A}
$$
$$
\frac{\Gamma \vdash e : \Diamond A \qquad x : A \vdash e' : \Diamond B}
     {\Gamma \vdash \mathrm{let\;val}(x) = e \;\mathrm{in}\; e' : \Diamond B}
$$

I changed the notation from $T$ to $\Diamond$, since this is actually the possibility modality of S4 modal logic! These modalities arise in applications like functional languages for distributed programming --- e.g., see Murphy et al's 2004 LICS paper ["A Symmetric Modal Lambda Calculus for Distributed Computing"][1].


  [1]: http://www.cs.cmu.edu/~concert/papers/lics2004/symmetric.pdf