Several answers have pointed out the affirmative reasons why dialetheism might be worth considering, implying that the chief motivation of dialetheism lies in the applicability to certain situations (whether they be logical or material) where the only correct description involves a dialetheia and which is otherwise intractable or has to be side-stepped and avoided.
However, it seems the chief motivation is of a negative kind: a (small) number of philosophers and logicians through the history of philosophy have found the original defense of the Law of Non-Contradiction (LNC) by Aristotle wanting.
As it was Aristotle who first introduced LNC, their first step is to reverse the burden of proof; it is a task of the defenders of LNC to give a theoretical justification - not for the unconvinced philosophers to justify their opposition to LNC. The 'opposition' in this step is simply the recognition that there is insufficient justification for holding LNC to be necessarily true.
Aristotle on LNC
Simply put, it is not clear what Aristotle exactly speaks about in Met.III when defending LNC. He mixes ontological, pragmatic, semantic and syntactic versions of LNC together. (Since there is not LateX support, I will just write the interpretations.)
It is not possible that the same object both possesses and lacks the same property.
No (rational) agent can simultaneously accept and reject the same sentence.
No sentence is both true and not true.
No sentence is both true and false.
A sentence and its negation cannot both be true.
Aristotle holds at one point or another that all these versions are transcendentally necessary and he ties them together as one principle. This SEP entry gives an overview on how Aristotle tried to tie these versions together and use them as a necessary condition for his ontological essentialism (i.e. his account of essence through the distinction between necessary and accidental properties).
His line of defense is the famous elenctic method. As the opponent who doubts LNC is not committed to non-contradiction, showing the opponent to be contradicting himself is not really a viable strategy. Instead, Aristotle tries to trick the opponent in showing the he accepts at least one instance of "x is F and is not at the same time not F", i.e. Aristotle's aim is to show that the opponent is committed to at least on thing that is not contradictory. He is thus arguing against trivialism, not modern dialetheism (which is not committed to the view that all contradictions are true, but only that some are).
Do you think that all the versions above are equivalent? That all can be defended in the same way? That one of the versions is analytically contained in another version? Aristotle did, and this was the status quo, including his arguments, until the early 20th century.
It seems to me not so difficult to imagine that some philosophers, starting with Jan Łukasiewicz, were not really impressed by this argument with heavy premises (Aristotelian essentialism!) and messy formulations. And, since logic was not seen anymore as laws of thought, and also not as correspondence with some metaphysical truth about how the world is, they started to think about how to deal with a logical possibility in which LNC doesn't necessarily hold (as Aristotle thought it would). At this point there are several possibilities to formulate weaker or stronger positions, and for the dialetheist the affirmative reasons above kick in, which lead them to take dialetheia seriously.
Allow me to draw a parallelism to the discovery of non-euclidean geometries. For centuries philosophers assumed this to be the only possible geometry. They adduced transcendental proofs (Kant tried to show that euclidean space is the "condition of possibility" to conceive of space), physical proofs (the physical space is just structured that way) and logical reductio ad absurdum-proofs (no other consistent geometry is possible). It was this last aim that actually got mathematicians like Saccheri to formulate, without intending to do so, non-euclidean geometries:
The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a reductio ad absurdum proof of any alternative to Euclid's parallel postulate. To do this he assumed that the parallel postulate was false, and attempted to derive a contradiction. Since Euclid's postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°.
The first led to the conclusion that straight lines are finite, contradicting Euclid's second postulate. So Saccheri correctly rejected it. However, today this principle is accepted as the basis of elliptic geometry, where both the second and fifth postulates are rejected.
The second possibility turned out to be harder to refute. In fact he was unable to derive a logical contradiction and instead derived many non-intuitive results; for example that triangles have a maximum finite area and that there is an absolute unit of length. He finally concluded that: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Today, his results are theorems of hyperbolic geometry.
... and that found some "nice applications" (though one could certainly argue that there are no logical reasons that compelled physicists to abandon euclidean geometry and we could have stuck with LET instead of SRT).
If you find this comparison misleading, there may be a more apt parallelism with the rise of multi-valued logics by giving up the law of bivalence.
The same happened with LNC. It was considered ontologically, pragmatically and logically necessary. Then it occurred, very late, that one could in fact construct logics weakening or abandoning LNC. From there these logics found some interesting applications in vagueness, paradoxa, etc. - an application which not everyone, as you show, finds compelling enough, because these applications are not logically compelling interpretations, and it is always possible to interpret them by maintaining LNC.