Isaiah 49:16 is waitnessing that God's people in this case Zion ( the choosen) are always in His sight, his eyes were ever upon them, and never withdrawn from them. Which shows how near and dear we are to him today as his choosen, what an affection he has for us, and care of us.
God, had drawn/carved or graven Jerusalem on his hands long before it was founded, and had it constantly before his eyes. The author's understanding of this is that God had laid out the plan of Jerusalem long before it was built, and that it was so dear to him that he had even engraven it on his hands. That is the same with us today. See Jeremiah 29:11.
The essential idea is, that Zion was dear to His heart; and that he had sketched it as an object in which he felt a deep interest, so deep to a point of carving its outlines on the palms of his hands, where it would be constantly before him. Think of a tattoo that reminds you of something dear to you. The author is not advocating tattoo piecing. However that is a topic for another day.
So what was God really saying to the Israelite children when they cried out to Him in despair? In their distress, they have accused Him of forsaking them and that He is not caring. He refers to the bond between mother and child, so sacred a bond. It is very rare for a good mother to forget her child.
Here God says to Israel that it could happen, that “But not so my bond with you,” Isaiah 49:15. God assures his choosen that he will never forget them. What an assurance. It is like music in their ears. How could I forget you?” see Isaiah 49:15b.
In His palm, God will hold them, He will carry them, He will lift them up lest they dash their feet against a stone. His eye is on them every moment of every day and every thought He has ever had toward them, every promise He has ever made, every prophecy that awaits fulfillment cries out to Him from the palm of that hand. See foot note.
Never will He forget those to whom He has pledged Himself. He has said it, it is fixed, it is carved, and it will be done. For those of us who love Him, there could be no greater words of consolation. Are there times you feel like God has forgotten you, like he does not care anymore. Know this today, that he loves you and has carved you in the palm of his hand.
You are encouraged to read the entire chapter Isaiah 49
Footnote: Henry Matthews commentary on Isaiah
work cited
Holy Bible- New King James Version
Commentary on Isaiah- Henry Matthews