To add a more conjectural interpretation to mwestwood's excellent and incisive reading of this scene, we might suggest that Daisy is a romantic almost as much as she is a materialist. It is true that her materialism and lack of inner-resources ultimately stand above her desire for romantic love, yet the fact that Daisy carries on a dalliance with Gatsby might be taken as a mark of her innocent yearning for ideal love.
Gatsby and Daisy fell in love before Gatsby went away to war and before Daisy was courted by Tom. Their story was interrupted - sundered - by material considerations. Gatsby was not wealthy enough for Daisy. But they loved each other.
When Daisy is confronted with proof that Gatsby now has achieved wealth beyond the basic standard she might have held as a young woman, she is also confronted with the idea that she could have had it all. She could have married the person she loved (Gatsby) and been wealthy too.
"[He] began throwing them, one by one, before us, shirts of sheer linen and thick silk and fine flannel.… While we admired he brought more and the soft rich heap mounted higher—shirts with stripes and scrolls and plaids in coral and apple-green and lavender and faint orange, with monograms of Indian blue. Suddenly … [Daisy] began to cry stormily."
The notion that both Gatsby and Daisy are pursuing a belated reconciliation as a way to recover an ideal romance is at the heart of their affair. This idea is also at the heart of Gatsby's character. He is the dreamer who believes beyond logic that he can repeat the past and - not only repeat it - make it better the second time around.
Daisy is not so sure. Her awareness of what she has lost in choosing Tom instead of Gatsby is poignantly conveyed by her response to the heaps of shirts in Gatsby's house.
“They’re such beautiful shirts,” she sobbed.… “It makes me sad because I’ve never seen such—such beautiful shirts before.”
In Daisy's mind, there is no way to undo what has been done. We can see this idea put to the test when Gatsby demands that Daisy say she never loved Tom. Daisy cannot bring herself to say this. She cannot bring herself to erase the importance of her past, however much she might desire a bright and romantic future with Gatsby.
No one, it would seem, is as capable as Gatsby in regard to insistent belief and purposeful faith in consummating a dream. Thus we might say that Daisy cries over the shirts because she realizes what she has lost - a chance at true romance. The dream, for Daisy, is now a strained with the melancholy of "what if." She carries on with Gatsby, but there is no way to recover those years that were lost and the innocence that was lost with them.
She cries perhaps because she knows there is no way to go back and choose Gatsby first instead of Tom.
The self-pity of such a reaction to a pile of shirts is well-aligned with Daisy's penchant for dramatics and her urge to be seen, adored and desired. Her egoism and materialism are both focused in this scene quite nicely. Clothes, a quintessential emblem of superficiality and surfaces, bring Daisy to a peak of emotion, all pointed inward toward an idea that she could have had more than what she ended up with.