Who between the two has a stronger immune system? A growing child 7-8 years old or an elderly person 75-80 years old?

The child should have a stronger system, if only because the older person's system has been used much more and is probably below its peak effectiveness.

An 80 year old is already knocking on deaths door.

An 8 year old has more resilience.

That is for the most part. It really depends on the individual.  

It' s not a skalar. Some parts work better in children, but the elderly probably have developed antibodies to several diseases

The child does. They don't have a big library of antibodies, but their T-cells can produce new antibodies at the drop of a hat. So they get sick a lot - like old folks - but they usually don't STAY sick. Old folks already have antibodies (and immunity) from many common diseases, but if they run into a new one their bodies are less able to produce a new form of antibodies as quickly as little kids. So the illness lingers, giving opportunistic infections or pre-existing conditions a chance to overcome the seniors' defenses.