How did Earth's atmospheric layers get their names?
Who gave the names Troposphere and Stratosphere

The stratosphere was firstly discovered by Léon-Philippe Teisserenc de Bort and Richard Aßmann (they did not cooperate) around 1900. Teisserenc de Bort denoted that layer as 'zone isotherme' in his publication on his discovery. However, he published in French. I remember to have read somewhere - not sure where - that he gave the two names 'troposphere' and 'stratosphere' to the two lower layers after discovering the existence of a quite distinct second layer. Unfortunately, I do not find the reference anymore. In the article Hoinka (1997) you will find some information about the discovery of the stratosphere.

What the Names mean

Troposphere

tropos (greek): turn(ing)

Turbulent mixing is relevant in this layer of the atmosphere => Turbulent Mixing Sphere => Troposphere

Stratosphere

stratum (latin): something which covers something else (pavement, blanket)

This layer of the atmosphere, which is not dominated by turbulent mixing, covers the lower turbulent layer => Covering Sphere => Stratosphere

Mesosphere

mesos (greek): middle
This part of the atmosphere is the third of five (from humans defined) layers of the atmosphere => Layer in the middle (not by distance/height but by counting layers) => Mesosphere

Thermosphere

thermos (greek): warm, heat

This layer can be heated to above 1000 °C by the sun. However, it does not feel 'warm' because of the low 'air' density. => Heated/hot layer => Thermosphere 

Exosphere

éxo (greek): outside, external

This layer is the most outward located layer of the atmosphere. => At the outer side located => Exosphere