# stored them in their hearts

"thought carefully about these matters" or "thought a lot about these events"

# What then will this child become?

Possible meanings are "What kind of great person will this baby grow up to be?" or "What a great man this child will become!" 

# the hand of the Lord was with him

The phrase "the hand of the Lord" refers to the Lord's power. Alternate translation: "the Lord's power was with him" or "the Lord was working in him powerfully"