Lord Krishna pick up the Govardhana Hill and saved Vrindavan from the wrath of Indra deva, But the use of an only little finger there is any specific reason?
Lord lifted the Govardhana Hill with one hand and not only with one finger, this is mentioned in SB 10.25.19:
ity uktvaikena hastena
dadhāra līlayā viṣṇuś
chatrākam iva bālakaḥ
Having said this, Lord Kṛṣṇa, who is Viṣṇu Himself, picked up Govardhana Hill with one hand and held it aloft just as easily as a child holds up a mushroom.
The commentary also says that
It is confirmed in the Hari-vaṁśa that Śrī Kṛṣṇa picked up the Govardhana Mountain with His left hand.