Suppose $M$ is a Riemannian 3-manifold. We introduce a function $t$ on $M$ such that the two dimensional surfaces "$t=\text{constant}$" in $M$ are nested topological 2-spheres with the innermost surface reducing to a point. For each value of $t$ let us assume that $S$ is one such surface and $\eta_a = \nabla_a t$ denotes the normal to $S$. The unit normal is then given by $n_a = (\eta \cdot \eta)^{-1/2}\eta_a$. Let $\xi^a:= un^a$ has the property that $\xi^a\nabla_a t = 1$.
I would like to show that in this case, the rate of change of any quantity related to $S$ with respect to $t$ is its Lie derivative by $\xi^a$.
All my efforts are failed, any help would be very much appreciated. Thanks.
From your comment I've determined that the precise statement you are looking for is $$f'(t(p)) = \mathcal L_\xi (f \circ t)(p)$$ where $f : \mathbb R \to \mathbb R$ and $p$ is any point on one of the surfaces $S_t$.
All we need is the property $dt(\xi) = \xi^a \nabla_a t = 1$ along with the chain rule: the Lie derivative of a scalar function is just the standard derivative of functions, so $$\mathcal L_\xi (f \circ t) (p) = d(f \circ t)_p(\xi_p) = df_{t(p)} \left( dt_p(\xi_p)\right)=f'(t(p))\cdot1.$$