What is the name of the property of viruses can activate a second time, with different symptoms?
Viral latency is the best term to use for viruses that can lie dormant.

I have not come across a specific term for latent viruses that recur with different symptoms. This is probably because "different" is very subjective. Many pathogens can cause several different forms of disease depending on factors such as the route of exposure (i.e. where and how the pathogen got to its destination). Obviously the latent infection has a different pathophysiology to chickenpox. Due to its latency in dorsal root ganglia, shingles typically results in a localised rash, but there are less common disseminated forms of shingles that can look very similar to chickenpox.