Which area of the brain is of particular importance for memory storage because it grows new neurons and makes new connections within the brain?

The answer is Hippocampus

It is found deep inside the medial temporal lob and is part of the overall Limbic system.

The Limbic system itself consists of the Hippocampus, thalamus,

cingulate gyru and the epithalamus all of which are associated with the processing of Memory.

Hippocampus itself is defined as an organ located just under the cerebral cortex.

It is particular good at managing long-term memory of a person. Hence, during Alzihemir's disease and dementia, Hippocampus is the first organ to get damaged.