Since summer is here, I will bore people a bit with my own ahistorical view
of Mordell. The main point is that Serge Lang's strategy, mentioned by several people, is
essentially correct. Nevertheless, we might add that

*you should believe the Mordell conjecture because most local
principal bundles should not extend to global principal bundles.*

I've spoken about this in one form or another many times, but
maybe this particular sentence hasn't received enough emphasis. So
I will explain it.

Jean-Benoit Bost has pointed out that this story is not entirely devoid of historical context, since Andre Weil's famous 1938 paper on vector bundles can be construed as giving motivation roughly of this nature.

Recall that Lang's idea is to consider the diagram

$$
\begin{array}{ccc}
X(F)&\longrightarrow & X(\mathbb{C}) \end{array}$$
$$\begin{array}{lcr}
\downarrow \ \ \ \ & & \ \ \ \ \downarrow\end{array}$$
$$\begin{array}{ccc}
J(F) &\longrightarrow & J(\mathbb{C})\\
\end{array}$$

for a smooth projective curve $X$ of genus $\geq 2$ with Jacobian
$J$ and an embedding $F\hookrightarrow \mathbb{C}$ of the number
field $F$ into the complex numbers. Lang suggested that

$$X(\mathbb{C})\cap J(F)\subset J(\mathbb{C})$$

should be finite.
This is indeed plausible and turns out to be true after Faltings
proof. Of course we are considering its status as motivation
rather than corollary, following the original question posted.

In fact, the plausibility is strengthened when we replace the diagram above by a refinement
$$\begin{array}{ccc}
X(F)\ \ \ \ \ \ \ \ \ \ &\longrightarrow &\ \ \ \ \ \ \ X(F_v) \end{array}$$
$$\begin{array}{ccc}
\downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ & &\ \ \ \ \ \ \ \ \ \ \ \downarrow j \end{array}$$
$$\begin{array}{ccc}
H^1_f(G_S, \pi_1(\bar{X}, b)) &\stackrel{loc}{\longrightarrow} &
H^1_f(G_v, \pi_1(\bar{X}, b))
\end{array}$$ and try to prove that
$$Im(j)\cap Im(loc) \subset H^1_f(G_v, \pi_1(\bar{X}, b))$$
is finite. Here, $$\pi_1(\bar{X}, b)$$ is some
$\mathbb{Q}_p$-algebraic fundamental group of $\bar{X}$ with
base-point $b\in X(F)$. The completion $F_v$ should be taken to
have degree one over $\mathbb{Q}_p$. The $H^1$'s are moduli spaces
of (locally constant) principal bundles for $\pi_1(\bar{X}, b)$, a
global one$$H^1_f(G_S, \pi_1(\bar{X}, b))$$ consisting of
principal bundles over some $Spec(O_F[1/S])$
and a local one
$$ H^1_f(G_v, \pi_1(\bar{X}, b)) \simeq \mathbb{A}^N_{\mathbb{Q}_p},$$
(almost naturally) isomorphic to affine space, consisting of
principal bundles on $Spec(F_v)$ (satisfying some technical
condition).

Thus, we are replacing $\mathbb{C}$ by a non-Archimedean
completion and the Jacobian (a moduli space of line bundles) by a
moduli space of principal bundles. The vertical maps assign to a
point $x$ the principal bundle of paths $$\pi_1(\bar{X};b,x).$$
This framework turns out to refine considerably the intuition that
$J(F)$ and $X(\mathbb{C})$ have very different natures inside
$J(\mathbb{C})$.

Now, *why should the Mordell conjecture be true?*

There are two steps.

A. The easy one: The map $j$, being a non-Archimedean period map,
is highly transcendental, and maps $X(F_v)$ to a Zariski-dense
compact analytic curve in $H^1_f(G_v, \pi_1(\bar{X}, b))$, which
therefore meets any proper subvariety in finitely many points. One
proves this by showing that certain transcendental functions on
$X(F_v)$ (the coordinates of the map) are algebraically
independent. Meanwhile, the localization map
$$ loc: H^1_f(G_S, \pi_1(\bar{X},
b))\longrightarrow H^1_f(G_v, \pi_1(\bar{X}, b)),$$ is algebraic,
and hence, has constructible image in the Zariski topology. These
are the different natures alluded to in the previous paragraph:
$$\begin{array}{ccccc}
H^1_f(G_S, \pi_1(\bar{X},b)) &\stackrel{\scriptstyle
\mbox{algebraic}}{\longrightarrow} & H^1_f(G_v, \pi_1(\bar{X},b))
& \stackrel{\scriptstyle
\mbox{dense analytic}}{\longleftarrow}& X(F_v)\end{array}$$

Therefore, it suffices to show:

B. The hard step: $Im(loc)$ is not Zariski dense, that is, the
localization map
is not dominant.

Now, why should this be true? Well, the moduli space
$$ H^1_f(G_v, \pi_1(\bar{X}, b))$$
consists of *local* principal bundles, while
$$ H^1_f(G_S, \pi_1(\bar{X}, b))$$
is a moduli space of *global* principal bundles. So it makes sense
that most local bundles should not extend to global ones when the
group $\pi_1(\bar{X}, b)$ is sufficiently large and non-abelian.
(Perhaps you will disagree...)

These thoughts were actually inspired by Yang-Mills theory: We
have something like local solutions to the Yang-Mills equation,
that is, on a small annulus (or a handle-body) embedded in a
Riemann surface (or a three-manifold). It seems they should not
all extend to global solutions. Natural enough, but quite hard to
prove in general. Galois cohomology, on the whole, appears to be
harder than Yang-Mills theory. One motivation for posting this
answer is the hope that some bright young person will have an idea.

Added: As mentioned by Matthew Emerton, the strategy outlined above is an extension of Chabuaty's method, which quite likely inspired Lang's conjecture as well (according to one reading of the notes to *Fundamentals of Diophantine Geometry*). There, the analogue of
$H^1_f(G_v, \pi_1(\bar{X},b))$ is
$$T_eJ(F_v),$$
the Lie algebra of the Jacobian, while the role of the global moduli space is played by
$$J(F)\otimes_{\mathbb{Z}}F_v.$$
My own feeling is that the 'local vs. global perspective' that emerges out of the principal bundle interpretation is somehow critical to understanding the Mordell conjecture, and constitutes a natural generalization not just of Chabauty's method, but of the arithmetic theory of curves of genus zero and one. In this sense, the essential motivation for Mordell's conjecture should not just be probabilistic, but something rather precise coming out of *class field theory.* It's fairly clear that this couldn't have been Mordell's reason for believing in it, but it is plausible, as mentioned above, that it was Weil's reason, in spite of his eventual non-committal assessment.