I have lightish olive skin and notice that I get darker area persisting even when my eczema has cleared. Why is this?

It sounds as though you have had a change in the pigment in your
skin. Episodes of inflammation in the skin from eczema and other
causes can lead to pigment changes, leaving the skin lighter or
darker. We call this process post-inflammatory hypo- (if lighter) or
hyper- (if darker) pigmentation. The darker your skin is to start
with, the more likely it is to get darker still. Except in a few cases,
the effect is not permanent, although it can last for several monthsor longer depending on how bad your eczema is. Please don’t try
treating these areas with any lightening creams you might see
advertised as these can be irritant to the skin, make your eczema
flare up and make the problem worse!