In Keats's poem "Ode on Melancholy', why are the trophies "cloudy"?

"His soul shall taste the sadness of her might,

And be among her cloudy trophies hung."

That's a great part of this poem by Keats.  The poem is really broken up into three parts (or stanzas, if it pleases you.)  In the first, the poet explains what not to do when you are feeling unhappy:

"NO, no! go not to Lethe, neither twist

Wolf's-bane, tight-rooted, for its poisonous wine;

Nor suffer thy pale forehead to be kist

By nightshade, ruby grape of Proserpine;

Make not your rosary of yew-berries,
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Nor let the beetle, nor the death-moth be

Your mournful Psyche, nor the downy owl

A partner in your sorrow's mysteries;

For shade to shade will come too drowsily,

And drown the wakeful anguish of the soul.

In essence, this part says "When you are depressed, don't try to ignore your depression (Lethe, the river of forgetfulness,) don't kill yourself (nightshade,) and don't become obsessed with symbols of death (beetle, moth.)

Stanza two discusses what you should do instead:

But when the melancholy fit shall fall

Sudden from heaven like a weeping cloud,

That fosters the droop-headed flowers all,

And hides the green hill in an April shroud;

Then glut thy sorrow on a morning rose,
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Or on the rainbow of the salt sand-wave,

Or on the wealth of globèd peonies;

Or if thy mistress some rich anger shows,

Emprison her soft hand, and let her rave,

And feed deep, deep upon her peerless eyes.

Here  the poet says, "when depressed, check out a pretty rose, or a rainbow, or beautiful peonies.  If someone is PO'ing you off, let them rant and rave, and see the beauty in her eyes."

Part three talks about the nature of happiness, that sorrow is always present:

She dwells with Beauty—Beauty that must die;

And Joy, whose hand is ever at his lips

Bidding adieu; and aching Pleasure nigh,

Turning to poison while the bee-mouth sips:

Ay, in the very temple of Delight
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Veil'd Melancholy has her sovran shrine,

Though seen of none save him whose strenuous tongue

Can burst Joy's grape against his palate fine;

His soul shall taste the sadness of her might, and be among her cloudy trophies hung

In essence, the poet is saying that beauty and joy are  great because they have a little sorrow in them.  Joy doesn't last, beauty must fade, and all that.  The wise person understands how the two are connected and can take both at the same time (burst Joy's grape, so to speak.)

To address your question directly,

And be among her cloudy trophies hung.

The point of this is that her trophies (think literally, like a bowling trophy) are not polished and shiny.  They are cloudy.  This goes along with the idea something great (the beautiful trophies) are not perfect.  There is a flaw (they are not polished.)  This goes along with the theme that the greatest pleasures are great because they have a bit of sadness in them.