They are unsullied by temptation.
One of the selling points (literally) of the Unsullied is their unfailing devotion. They will not betray the one they serve, not for money or love or greed or lust. Presumably this is what the name refers to. The sullying of their manliness, or the possible sullying of their souls by removing instincts such as mercy or compassion during the training process which involves murdering babies and their own pet dogs, is unlikely to be something their trainers or owners care about. The name is chosen to reflect their positive aspects (from the point of view of those selling them), not anything more human.
"In Yunkai and Meereen, eunuchs are often made by removing a boy's testicles, but leaving the penis. Such a creature is infertile, yet often still capable of erection. Only trouble can come of this. We remove the penis as well, leaving nothing. The Unsullied are the purest creatures on the earth." He gave Dany and Arstan another of his broad white smiles. "I have heard that in the Sunset Kingdoms men take solemn vows to keep chaste and father no children, but live only for their duty. Is it not so?"
"It is," Arstan said, when the question was put. "There are many such orders. The maesters of the Citadel, the septons and septas who serve the Seven, the silent sisters of the dead, the Kingsguard and the Night's Watch . . ."
"Poor things," growled the slaver, after the translation. "Men were not made to live thus. Their days are a torment of temptation, any fool must see, and no doubt most succumb to their baser selves. Not so our Unsullied. They are wed to their swords in a way that your Sworn Brothers cannot hope to match. No woman can ever tempt them, nor any man."
His girl conveyed the essence of his speech, more politely. "There are other ways to tempt men, besides the flesh," Arstan Whitebeard objected, when she was done.
"Men, yes, but not Unsullied. Plunder interests them no more than rape. They own nothing but their weapons. We do not even permit them names."
-- A Storm of Swords, chapter Daenerys II