Added 22, October:

While reflecting on this question and the subsequent discussion, I fell again into a
bit of a sermonizing mood. I hope I will be forgiven for inserting two words of
caution.

There is a difference, probably significant, between lack of interest
and *aggressive* lack of interest. Most people will know what I mean by this,
I think. The latter is probably best avoided. Now, I wouldn't be too strong about
doing away with
it altogether since some people do seem to derive substantial creative
energy by being *against* something, both in mathematics and in the world at large.
But for most of us,
the less contemplative version of disinterest is, I believe, more destructive than
constructive.

Since I've already expounded on the mainstream view,
it might be alright to vary on it a bit.
It's good to educate oneself about fashion and probably sensible to
follow it a fair amount. But then there are the true cliches about
independent thinking.
I'm old enough to have witnessed first-hand the fervor surrounding the
wonderful proof of
FLT. At this point, I'm sure many of us can recite by heart all the reasons it is more important
than Goldbach, for example, and the significance of the relation to modularity, etc.
This may indeed be a reasonable point of view.
However, to be honest, I rarely got the impression then that the young expert in the common room was
doing more than just that: reciting the viewpoint.
I suppose I'm just repeating the platitude that fashion might be quite
sensible, but slavish adherence to it is not. So if you feel strongly about
some specific Diophantine equation that doesn't quite fit the standing paradigms,
my own advice is to to think about it frequently enough to see if some real ideas develop. I hope I'm not misrepresenting him, but Swinnerton-Dyer once claimed that very few people were interested in L-functions around the time he was first experimenting with points on elliptic curves. Even now, he will speak with considerable passion about a single equation, or at least, about a single special family.
(Nonetheless, I hope these paragraphs don't strike you as regurgitation of some superficial
faith in 'diversity'. I have some of that as well, but some mathematics is clearly better than others.)

Regarding Goldbach, I have the curious impression that it's about to gain substantially in respectability,
especially with the remarkable ascent of additive number theory related to the work
of Gowers, Green-Tao, et. al. I try to think about it myself
every now and then, partly because of the influence of Shinichi Mochizuki,
who insists on viewing the connection between additive and multiplicative structures in arithmetic
through the prism of non-abelian fundamental groups.

Oops! I'd better clarify right away that my remarks on Fermat and Goldbach are meant in no way as criticism of Jordan's nice answer.

Original answer:

A proper answer might require at least an essay, but here is an abridged attempt.

Two classes of equations have already been discussed in the other answers:

(1) Some equations are 'just interesting' for their special or exotic properties. I quite
like the classical mathematics generated by the equation $$x^3+y^5=z^2$$ mentioned in Mike Bennett's comment. Smooth cubic surfaces like
$$x^3+y^3+z^3+w^3=0$$
are also nice with their twenty-seven lines that eventually do influence their arithmetic. Fermat equations might be appreciated for their similarly high degree of symmetry that induces the complex multiplication on their Jacobians. By the way, regardless of their age, I find Calabi-yau varieties quite fascinating myself, since the interplay between Hodge-theoretic and Diophantine properties
is a subtle phenomenon deserving of study.

There is obviously no objective criterion being offered here, but still an equation may appear as especially interesting in the same way certain spaces are interesting or some animals are interesting.
They excite a certain desire to know about them in considerable detail. Investigation with affection is usually richly rewarded in these cases.

(2) Equations that come up while studying some other problem. Pell's equations were mentioned above, and one could consider other norm equations while studying number fields. Keith mentioned also the relation between the far-reaching ABC conjecture and Mordell's equation. One other nice class of elliptic curves of this nature are $$y^2=x^3-n^2x,$$ which were famously connected to the congruence number problem on the area of right angle triangles with rational sides and areas. A spectacular example in the ABC vein is Mazur's study of points on modular curves that gave rise to uniform bounds on the torsion subgroup of *all* elliptic curves over $\mathbb{Q}$. And then, integral points on Siegel moduli spaces were bounded by Faltings in his proof of the Mordell conjecture. In short, one can even apply one kind of equation to the study of another (family).
In any case, for this class, one presumes the equations are as interesting as the motivating problem.

However, the perspective I actually wished to mention sidesteps the question somewhat. The view is perhaps the most mainstream and reactionary possible in this context, but closest to mathematical practice as I see it. It says most equations have or lack interest not in and of themselves. Rather, the main issue is the questions we ask about them. I will remind you of three examples:

(i) Consider the various conjectures on $L$-functions. Say the conjectures of Birch and Swinnerton-Dyer. To oversimplify the case a bit, suppose you could prove the conjecture in full up to the last detail for the single elliptic curve
$$498208y^2=x^3+309208472x^2+1204948278x+3920984$$
or with some other choice of coefficients as random as you want. There will be little disagreement that this would be highly interesting.

In case you think that elliptic curves are already deserving of
special attention, choose a random collection of homogeneous equations
$$f_1=0, f_2=0, \ldots, f_n=0$$
in $m$ variables. Most of the time, they will define a smooth projective variety $X$. Bloch and Beilinson have conjectured that the order of vanishing of $L(H^{2i-1}(X),s)$ at $s=i$ is equal to the rank of the Chow group of algebraic cycles of codimension $i$ homologically equivalent to zero. Being able to prove that statement for any given $X$ chosen randomly would be highly interesting. Of course because one expects this to be so difficult, people concentrate rather on special $X$'s. [In case you are wondering about their relevance, algebraic cycles on $X$ should rightly be thought of as 'generalized solutions'.]

(ii) Continuing with the same notation, suppose $X$ happens to be a Fano variety, which will often happen if
the degrees of the polynomials add up to something rather small compared to the number of variables. In that case,
Manin has conjectured that the rational solutions in some fixed number field $F$ (depending on the equations) will be Zariski dense. That is, this is a set of equations with very simple geometry, because of which it should be consistently easy to find many, many solutions, as soon as some obvious obstruction is overcome.
Once again, you are free to attempt this after choosing the $f_i$'s in as arbitrary and as unaesthetic a manner
as possible. As with the Beilinson-Bloch conjecture, the more random your choice is, the more impressive your
result will be, in some sense.

(iii) Close to my own heart is the effective Mordell conjecture, which asks for an algorithm to find all rational
solutions to a generic equation
$$f(x,y)=0$$
with degree at least 4. As a consequence of the fact that such an algorithm is unknown, it becomes of considerable interest to
be able to list full solution sets in any given case. Sometimes, it's easy to show that there are none, such as
$$x^4+y^4=-1,$$
just to be absurdly simple. However, once such silly reasons for triviality are excluded, for example, if you happen to notice already one
solution, it is notoriously difficult to list the whole solution set. Here is an example due to Bjorn Poonen:
$$y^2 = x^6 - 2x^4 + 2x^3 + 5x^2 + 2x + 1.$$
You will easily see the solutions $(0,\pm 1)$. However, it requires quite a bit of technology to show that
$$(0,\pm 1), (-1,\pm 1), (1,\pm 3)$$
is the full solution set. You can see that this particular equation does seem pretty random. On other hand, because of an enduring
focus on the difficult and structurally demanding question of an algorithm, any example of this sort generates quite a bit of interest.

Many people have suggested that the variety of *techniques* that come up in attacking a problem are as important as the problem
itself. There is something to this, in as much as we would like the problem to tell us as much as possible about
the mathematical landscape in general, which is, after all, the ultimate object of our investigation. On the other hand,
once certain overarching
questions have already been established as powerful probes for this process, being able resolve them for any specific
object is interesting regardless of how pretty or ugly someone may find the object on its own. Obviously, this
is *the* raison d'etre for good conjectures.

Added 26, October:

Eventually, I stumbled on to the 'box equation' I referred to in the comments. It is
$$a_1^2+a_2^2=b_3^2;$$
$$a_1^2+a_3^2=b_2^2;$$
$$a_2^2+a_3^2=b_1^2;$$
$$a_1^2+a_2^2+a_3^2=c^2;$$
defining a surface in $\mathbb{P}^6$. A rational solution with $a_1a_2a_3\neq 0$ corresponds to a 'rational box' having all edges, face diagonals, and long diagonal rational. Apparently, the existence of such a thing is still unknown. There is a nice discussion in this
paper of Stoll and Testa. Of course, you have to decide for yourself if it's interesting. The flavor of it is somewhat reminiscent of the congruent number problem, and I think that was why it caught my attention. That is, given my own bias, I had to consider for a minute or two if there were a sneaky connection to a 'conceptually sophisticated' problem. Stoll and Testa relate it, in fact, to the Bombieri-Lang conjecture.