This probably means that the parents may choose the name based on something important to them. For example, my friend was named for his uncle, who died when my friend's father was 16 years old (the uncle was 20 years old). To my friend's father, this name was very important. It could also mean that a name will influence the person throughout his or her life as Alexandre said.

These names can have great influence on either the person the name was given to, or people whom hear the name. For example, I was named after my uncle. I feel that I have a great responsibility to make my uncle and parents proud. My name holds "psychological weight". Alternatively, a viking name could be "Victor the Bone Crusher". If I met this hypothetical viking I would be terrified. Another man (or woman) can have a word that sounds strong or too weak can hold "psychological weight".

It means your name has influence in your psychology as a human.