The righteous who walks in his integrity -
blessed are his children after him!
Blessed are his children who grow up in a home under his influence. They have a father who models the righteous life before them. They have a father who instructs them in the way of righteousness. They have a father who shows them steadfast love and faithfulness, who loves his wife, and who fears the Lord. For all of this is to walk in integrity.
It is error to restrict integrity to being honest and even to following one's principles. Such "integrity" is defined as being true to oneself. What matters is to be true before the Lord. Honesty that shows no regard for love of one's neighbor is sin. It is a misuse of true honesty. And it is sinful to follow one's principles if they are not God's principles.
A man who will not cheat a client, but who will not love his children, is not walking in integrity. A man who leaves a legacy of being ruthlessly honest and no concern for encouraging others is not walking in integrity. However much money he may leave his children due to his integrity in business, he has not shown integrity before his children and thus leaves them poor in spirit. And if he has left them a legacy of strict law abiding with no understanding of mercy and grace, he has failed to walk in integrity.
Bless your child by integrity that is marked by the fear of the Lord and the love of the Lord; by integrity that nurtures your children and bears the mark of Christ, which is love for one another.