# I will fasten him, a peg in a secure place

Yahweh causing Eliakim's authority to be strong and secure in the king's palace is spoken of as if Eliakim were a peg and Yahweh will set him firmly in the palace wall. 

# he will become a seat of glory for his father's house

"Eliakim will bring honor to his family" 

# his father's house

"his father's family" or "his family"