So my actual favorite first example is to do this with Fibonacci numbers as a linear recurrence relation, but that's not really a "physical" interpretation. Let me give you my favorite physical one:

The essence of special relativity is that acceleration is a bit weirder than you think. In particular when you accelerate by amount a in some direction x, even after accounting for the usual Doppler shifts you will find that clocks separated from you by that coordinate, appear to tick at the rate 1 + a x/c² seconds per second, where c² is a fundamental constant. Clocks ahead of you tick faster, clocks behind you tick slower (and indeed appear to slow down and approach a ‘wall of death,’ more technically called an ‘event horizon,’ at a distance c²/a. (This effect is called the ‘relativity of simultaneity,’ and it is in some sense the only real prediction of special relativity, as the rest of this comment will show—the other effects of ‘time dilation’ and ‘length contraction’ are second-order and can be derived from this first-order effect.)

This means that the transformation equations for moving into a neighboring reference frame are not the ones that Galileo and Newton proposed,

t' = t
x' = x – v t

but slightly modified to (to first order in v, so only considering small velocity changes)

t' = t – (v/c²) x
x' = x – v t

where w = c t is a measure of time in units of distance using this fundamental constant. How do we generalize and get the full solution? We can do it by looking in the eigenvector basis. Consider new coordinates p = x – c t and q = x + c t, given any (x, t) you can find a unique (p, q) which describes it and if you want to get back those values you would say x = (p + q)/2, t = (q – p)/(2 c). But feed these magical coordinates that come from eigenvectors into the above transform and it "diagonalizes",

p' = (1 + v/c) p
q' = (1 – v/c) q

and therefore if you want to make a big change in "velocity" c φ (here instead φ turns out to be "rapidity") out of N smaller changes, you can repeat this transform N times with little boosts by v/c = φ/N, and you will stitch together the full Lorentz transform out of little first-order Lorentz transforms:

p' = (1 + φ/N)^N p = e^φ p
q' = (1 – φ/N)^N q = e^{-φ} q

Transforming back and using the hyperbolic sine and cosine, sinh(x) = (e^x – e^{-x})/2, cosh(x) = (e^x + e^{-x})/2, the full formula is

w' = w cosh(φ) – x sinh(φ)
x' = x cosh(φ) – w sinh(φ)

where w = c t is a simple time-in-units-of-meters coordinate. Usually we denote cosh(φ) = γ, sinh(φ) = γ β, which gives this the more familiar form you'll find in textbooks, and the identity cosh²x = 1 + sin²x gives a formula γ = 1/√(1 – β²) for the latter... but this ‘rapidity form’ is in some ways more elegant. Anyway, point stands, from the "first-order" transform you can derive the "full" transform just by building any large velocity change out of an infinite number of infinitesimal velocity changes, and this is the source of the factor γ which describes time dilation and length contraction.

Okay, now for physical interpretation. You asked what physical meaning these eigenvalues and eigenvectors of the Lorentz transformation have, and the answer is this: the eigenvalues (1, 1) and (1, -1) of the Lorentz matrix represent light rays, the p/q description we came up with above was a description of spacetime in terms of light-ray coordinates where we identify an event at a particular place and time with the light rays that it casts, announcing that the event has happened, in the +x and -x directions. On the negative side, these are also the last light rays that were able to touch the event before it happened, so represent "everything it could have possibly known about" -- there is a space between these two "light cones" which is its "relativistic present," the things that anything which was there at the event cannot know about until the future.

The eigenvalues, exp(φ) = sinh(φ) + cosh(φ) = γ + γ β = √[(1 + β)/(1 – β)] and exp(-φ) = √[(1 – β)/(1 + β)], are the Relativistic Doppler shifts of those light rays. Indeed one can read them as e.g. exp(-φ) = 1/γ * 1/(1 + β) , here 1/(1 + β) is the standard Doppler shift formula from nonrelativistic physics and 1/γ is the decrease in frequency due to time dilation.