The baby might be very inquisitive. With Athena as its mother, it would inherit a thirst for knowledge. It could constantly be asking questions, eager to learn about the world around it, whether it's the secrets of the ocean (from Percy's side) or the wisdom of the heavens.
Given the combination of Percy's impulsive nature and Athena's calculated wisdom, the baby could have a very interesting personality. Sometimes it might act on impulse, like jumping into a situation without thinking too much. But then, Athena's influence would kick in, and it would quickly analyze the situation and find a more intelligent way to deal with it.