I did it and this is what I learnt.
Crash course of calendar history ...
To understand the origin, the etymology of these names, we have first to refer to Latin and to Roman culture.
Indeed Latin names of months haven't changed since 8 AD!
I will happily omit all changes, over the centuries, in the definition of months, I will consider only the names.
If you want to know more, please check references.
In Latin, the 12 Julian (and Gregorian) months are:
Iulius (formerly Quintilis)
Augustus (formerly Sextilis)
as you see, many of them sound quite familiar to their English, German or Italian versions!
At the beginning, there was the Ancient Roman calendar, introduced in 753 BC, according to tradition, by Romulus, the founder of ROME.
It had only 304 days, it began in spring and - the most important thing for this post - it had just 10 months!
Martius, Aprilis, Maius, Iunis, Quintilis, Sextilies, September, October, November, December.
This calendar was reformed in 713 B.C. by Numa Pompilius who introduced two extra months:
Ianuarius and Februarius
and he kept all other names. The Numa's calendar had 355 days.
The next reform was performed by Julius Cesar in 43 BC. This new calendar (called Julian Calendar) had 365 days and 366 in the leap year that came every four year.
After Julius Cesar's death, Quintilis was renamed Julius.
In 8 AD, the Roman Senate operated another change: Sextilis became Augustus (Read below!)
In 1582, Pope Gregory XIII made the last and most recent modification, however this time the change affected mainly the calculation of the leap year, the names of the months were held.
The "new" calendar, that by the way is the one we are currently using, was named Gregorian calendar.
For the record, the new rule for the calculation of the leap year were introduced to better match the calendar and the revolution time of the Earth around the Sun: the Julian calendar lags 11 minutes every year (or a day every 128 years), while the Gregorian one only 26 seconds (or a day every 3323 years).
... and now the etymologies!
Ianuarius: dedicated to Janus, the Roman god of beginning and transitions.
He is two-faced since he looks both into past and future. This is also the reason why the first month of year, a transition period between old and new, is dedicated to this god.
Fascinanting, isn't it?
(Janus, Vatican Museums, from Wikipedia)
Februarius: from februum (="purification") because in this month took place the purification ritual called Februa.
Martius: from Mars the Roman god of war.
In March the spring arrives and it is the perfect time to start a war (at least according to the Romans).
Incidentally, Martius signed also the beginning of the new year in the ancient Roman calendar.
The origin is uncertain but there two main theories.
It could refer to the Roman goddess Venus and in this case Aprilis is based upon Apru i.e. the Etruscan borrowing of Greek Aphros. Venus or Aphrodite was goddess of love and beauty.
Another possible etymology and perhaps the most traditional one, is from the verb "aperire" meaning "to open", a clear reference to the blossom of plants and flowers in springtime.
(Birth of Venus, Sandro Botticelli)
Maius: dedicated to Maia (="the great one") goddess of fertily. And her festival was held in May.
Iunius: from the Roman goddess Juno, corresponding to Greek Hera. She was sister and wife of Jupiter.
Ovid, as in the case of Maius and Iunius, provides two another etymologies. Iunius comes from the Latin word iuniores, meaning the "younger ones" as opposed to maiores ("elders") for which the preceding month Maius was named.
Quintilis means "fifth month of the year" (quinque= "five").
The number "five" and the name quintilis were a clear reference to the position of this month in the ancient Roman calendar even if, since 713 BC, it became the seventh month!.
The lastest name Iulius was decided to honor Julius Cesar who reformed the Roman Calendar in 43 B.C.
I guess he deserved it.
Sextilis from sex (="six"), it means "sixth month of the year". Once again a reference to the ancient Roman calendar of 10 months.
The new name "Augustus" was adopted in 8 a.D. to honor imperor Augustus, who, according to Macrobius, chose this month because it was the time of several of its triumphs including the conquest of Egypt. As case of the life, Augustus also died in August, precisely on August, 19th, 18AD.
September: it means seventh month of the year (septem="seven"). Once again a reference to the Ancient Roman calendar.
October: from octo(="eight"), meaning "eighth month of the year". Read above.
November: from "novem" (=nine), "ninth month of the year". Read above.
December: from "decem" (="ten"), "tenth month of the year". Read above.
In conclusion, 8 month names derives from gods, festivals or VIP (=Julius Cesar and Augustus) while the last four reflect the original position in the ancient Roman caledar.
Obviously, the etymologies I outlined here only applies to those languages in which the month names have similarities with Latin names. When this is not the case (e.g. Swedish, Polish), the etymologies are consequently different!