"I only care about honoring her memory." I only want her memory (Her past) to be respected and esteemed and not to be forgotten. Also to be remembered by her goodness. And you can also Honor someone's memory even if they're alive like when someone moves from a city to another city or to another country. Their friends, family and so may honor their memory as an expression of missing and respecting them.

When someone is alive you can honor them directly. When they are dead you honor their memory.

"B" loved and respected his mother very much, and it is very important to him that other people remember his mother in a very good way.