Many other answers mentioned ODE, but the discrete version seems simpler to introduce at an early stage. One could use system of sequences to show a magic trick and then explain it with eigenvectors, it might give both intuition and motivation. Let me be more specific.
Example problem
Suppose we have two sequences $(u_n)$ and $(v_n)$ such that for all $n\ge0$:
$$\begin{cases} u_{n+1}=3 u_n+v_n \\ v_{n+1} = 2u_n+2 v_n\end{cases}$$
( if needed insert any modelling motivation, such as evolution of a population of insects with young ones and adults, and adjust the parameters to make it somewhat realistic.)
How does the sequences $(u_n)$ and $(v_n)$ evolve? Can we give exact expressions for them?
Side note: before going on, at this point I would discuss what can easily be said without computing anything: if $u_0$ and $v_0$ are positive then the sequences are increasing and tend to $\infty$ for example.
Magic trick
Let us define for all $n\in\mathbb{N}$: $a_n=2 u_n+v_n$ and $b_n=u_n-v_n$. Then observe that
$$a_{n+1} = 2 u_{n+1} + v_{n+1} = 8 u_n+4 v_n = 4 a_n$$
and similarly $b_{n+1}=b_n$. So, $a_n=4^n (2u_0+v_0)$ and $b_n=u_0-v_0$. The two recursions have been decoupled! Then by solving a system we get
$$\begin{cases}
u_n = 4^n(\frac23 u_0+\frac13 v_0) + \frac13(u_0-v_0)\\
v_n = 4^n(\frac23 u_0+\frac13 v_0) -\frac23(u_0-v_0)
\end{cases}$$
We can now answer far more questions on the system easily (for which values of $u_0,v_0$ does $u_n$ go to infinity? Does $u_n$ grow much faster than $v_n$? etc. - one can ask this questions beforehand for more impact) But to students it should look like a magic trick: how could one think of introducing precisely $(a_n)$ and $(b_n)$ as above?
To the matter
Then one should be in a good shape to explain eigenvectors and eigenvalues, by rewriting the system as a vectorial sequence with a recursive property of the form $U_{n+1} = A U_n$ where $A$ is a matrix. The case of diagonal matrices is easy, the whole point of eigenvector being (at this point) to diagonalize a matrix. Alternatively, one can dissect the trick, and look at how one should choose the coefficients in $(a_n)$ and $(b_n)$ to make it work. Then eigenvectors and eigenvalues show up (but one has to introduce things in matrix form at some point, of course).