An invitation is most often thought of as a written or oral request to appear at an event such as a wedding, party, dinner, etc. But it is also anything that causes you to be attracted or allured. "Her sheer beauty was an invitation to talk." "The intellectual challenge of nuclear physics was all the invitation she needed.". Hold your head up high = be proud of who you are, where you come from and what you do, especially in challenging times So the author is saying that his children have heard him say "hold your head up high" many times and that they have been attracted to that thought and are proud of who they are. This is something that his mother taught him and which he values and views as a gift to pass on to his children.

Her monther told her to "hold you head up high" to face the world... to face her adversity... to be confident, and to not hide her blind eye and scared face... Tough not so extreme as the author's eye/face, everyone has to have something deep in one's heart that he wants to hide and avoid... I believe the author didn't only shout the words out to her children, yet she let them fell the way.....in everyday life, by passing along the love and encouragement her mother gave her... otherwise it would be too thin to express the feelings... and the article would be less beautiful ...

I believed, it's here means “temptation” or "suggestion" or "encouragement" You may noticed that "my children has FELT its invitation" And it's "The gift my mama gave me lives ON in another generation" a very good article!!!

probably it means ""deep meaning"