Presumably, Eru didn't make him evil, rather it seems like he became evil from spending too much time in the void. ("if you gaze too long into the abyss...")

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It would hold that logically, if Manwe, Namo, Yavanna, etc all had various purposes, then Melkor should have had a mandate from Eru as well, likely related to Manwe's in some fashion. If Sauron is any example, this mandate and his own misunderstanding of it lead to his fall.

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So, I'm guessing Melkor's purpose was to represent the concept of the finite, death, and ending of things. This is in opposition to Manwe's purpose, to represent the infinite, eternal, and new beginnings. It also helps explain why Melkor was so powerful relative to the other Valar, he understood their limitations and consequently was the second best at pretty much every other concept, with the exception of some of Manwe's (which would be infinite, but Manwe's understanding of course limited itself)

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Melkor could (and did) tear down everyone else's creations. The deep irony would be that Melkor, for all of his rebellion against Eru, was literally doing what he should have done anyway, which is reflected in the line Eru tells him. Had he been less of a dick about it, the other Valar would have had a much greater understanding of both him and themselves.

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His chief failure was against the Children of Eru, by turning Elves into Orcs and shadowing the gift of men (which was particularly bad because, as the Vala of Death, he alone would have understood their purpose and gift).

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EDIT - Flushing out theory more and responding to some comments --------------------------------------------------

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Neither Melkor nor any finite mind could be the equal (or even close) to Eru. However - based on this theory Melkor is the best finite mind capable of contending with Eru's infinite nature by understanding the limitations of Ea. (Granted this is of course only because Eru himself put finite boundaries on Ea)

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Manwe was closest to Eru's heart because he was a finite representation of the infinite concepts Eru represented. Eru of course cared for Melkor (and very very likely still cares). Melkor is in many ways an opposite of Eru, necessary but not inherently evil (as nothing is). Melkor can do nothing to sully Eru's work, even his destructiveness serves Eru's purpose. That being said, Melkor's evil is manifest in some of his actions and good that he could of done which was lost, such as Orc creation and Men not fully understanding the gift of men. I would argue the majority of Melkor's evil comes from what he hasn't done, not the acts he has done.

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Melkor's mandate was help all other beings bring forth new works by destroying their old works and making room for new creations. In many ways despite being the most powerful of the Ainur he was just a very necessary garbage control system, and should have had a lot more humility.

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He destroyed the Two Trees before their time, weakening both himself and the other Valar (as well as not preserving enough of their light for another incarnation), but more importantly didn't continuously recycle the world, nor share his knowledge of intrinsic change with the other Valar or the Children of Eru. This led to Manwe trying to 'save' the world in Aman, leading both to a beautiful but hopelessly static continent, and a continent overwhelmed with destructive change. It's likely that a good Melkor would have given the Elves a better understanding of change.

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Had he accepted the guidance of Manwe I'm sure he'd still be disliked by some of the more naive children of Illuvatar, but Manwe, Namo, and Nienna would have understood his necessary functions. (Nienna in particular) Its also possible that Eru would have worked with him to create Mortal races (I'd like to think Eru wouldn't saddle a being with the responsibility of killing stuff without likewise being able to participate in creating it, which is possibly why Melkor had this 'authority' to create sentient beings (although only with the approval of Eru, as with any sentient being)), and that the Dwarves are a substitute for this action which did not take place on account of him turning evil. (Had he created good sentient beings with Eru, Aule might have had someone to teach)

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Im guessing since Manwe was air, Ulmo water, and Aule earth, Melkor was fire, which of course is a necessary "reforging" in a finite world with finite resources.

The balrogs were probably his "people" as Orolin was Manwe's and Osse was Ulmo's. It explains why they were so stupid and incapable of initiative, spirits of destruction and renewal robbed of the creative purpose behind their destructive acts.

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So what do you think? Any glaring holes or deep problems with the theory?