Alveolus (pl. alveoli) is derived from the Latin alvus, the diminunitive form of alveus (a channel or cavity). Thus the term is used generally to describe small structures that have a hollowed-out form.
The word was first used in anatomy by Vesalius around 1550 to describe tooth sockets in the mandible and maxillae; for over 300 years this was the only anatomical definition. The French anatomist Rossignol, in 1846, was the first to describe the microscopic air sacs in the lung as alveoli, probably because magnified sections of lung tissue reminded him of a honeycomb (in Latin, the cells of honeycombs were called alveoli, a definition that still holds forth in modern-day English).
The latin root alveus has entered English as well, meaning "river channel". It also is used in anatomy to describe a tract that runs much of the length of the hippocampus and forms the floor of the "channel" that is the temporal horn of the lateral ventricle.
The most recent anatomical use of alveoli, stemming from the latter 19th century, is as an alternative (and superfluous) name for the acini of exocrine glands.
Honeycomb-like alveoli in a cat lung as drawn by Rossignol
from Principles of Human Physiology (1860)
by William Benjamin Carpenter