The spell mage hand has no visual presence except that the affected object moves. Is it possible to employ the spell mage hand to make a Sleight of Hand skill check to take something unnoticed? Can an eidolon make such a check if it has the evolution basic magic?
There is a Prestige Class (Arcane Trickster) that specifically has that power as the first ability you get from the class, and it even increases the DC by 5 when doing it from a distance.
So No, you cannot. The spell doesn't give you enough precision to steal pockets, it will only do what the spell description says, because regardless of the name, it doesn't actually create an invisible extra flying hand.
You could, however, steal something and then make it fly away from you or your target. As long as he is distracted, he won't notice it. But keep in mind that Mage Hand has Vocal (you must speak) and Somatic (hand gestures) components, so your target is allowed a Spellcraft check to figure out what you are doing.
Target one non-magical, unattended object weighing up to 5 lbs.
(mage hand spell description, emphasis mine)
The usage of Sleight of Hand to take something unnoticed is specifically to take something from someone. If they have it, it is not unattended, and thus it is not a valid target for mage hand. As mentioned, the arcane trickster prestige class is one way to overcome that limitation.