Modern human beings, especially women, cut their armpit hair. It seems to me the armpit hair is trivial/useless. Shortly speaking, what is the armpit hair for?
I have been trying to read up a bit on this. I started out looking on wikipedia and it seems there are three hypotheses which are not necessarily mutually exclusive.
1.Aid the wicking of sweat away from the skin
2.Reduce friction between the thorax and upper arm
3.Facilitate the release of sex pheromones
The first would seem possible because it would help to prevent over-heating of the lymph nodes under the armpits and that also fits with the fact there are a lot of sweat glands there.
The second, I can't find any citations which support this but I understand that people who shave their armpits tend to suffer from dry underarms. This could be a result of increased friction.
The third is apparently controversial although it seems perfectly reasonable and highly likely to me. There is a famous experiment by Claus Wedekind who got men to wear a t-shirt for a few days. He then got women to choose a most preferred male purely by the scent of the t-shirt and found interesting results relating to the MHC profiles. The armpit hair could help to trap the odour produced by the sweat and help with mate attraction, like some kind of scent sponge. I don't understand why this is so controversial, loads of animals influence mate choice with scent - perhaps people don't like to think we are genetically attracted to each other by sweat because it seems animalistic and less romantic.
In summary, there is no sure answer and it could well be all of the above, or something entirely different.
It's not proper to think of evolution in teleological (purpose-driven) terms--armpit hair is probably a trait that was never eliminated because it is not harmful to the species, rather than a useful trait that was specifically "selected for".
Microbiome manipulation is a common theme in these answers/comments, and there are experimental methodologies for investigating such things. For example: "Topographical and Temporal Diversity of the Human Skin Microbiome" at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2805064.
One selection pressure that hasn't been mentioned is the reduction of harmful bacteria that occurs when providing a growth advantage to more benign bacteria. Many hairy areas are those where bacteria in general can live and prosper. Various ways hair could impact the microbioime are are easily imagined, for example greatly increased surface area should favor aerobic vs anaerobic bacteria. More specific host-bacteria interaction, e.g. specific bacteria being selected for due to the presence of certain hair surface proteins or the microstructure of the hairs surface might be harder to investigate. How hair surface varies in different location might be informative.
The explanation I heard from the documentary called "Curiosity: World's Dirtiest Man" is that armpit hairs allows for the growth of microbes which allow for the production of smell. This smell is useful for the attraction of mates in primates (although this function is probably lost in humans as our society became more hygienic).