From
http://www.axa.com/navigation/nav/mainframe.asp?id=7 and then click on each year of history:
In 1816,
Mutuelle de l'Assurance contre l'incendie is founded in Paris. The following year,
Ancienne Mutuelle begins operations. In 1978 Ancienne Mutuelle changes its name to
Mutuelles Unies. Beginning in 1982: the pace of external growth begins to accelerate. Mutuelles Unies aligns with
Drouot, strengthening its positions in the life and property-casualty insurance markets. 1985: A consensus is reached: the Group needs a shorter
and more upbeat name. It should start with an "A" so that the Group is first inlists and directories, and it should be easy to pronounce in every language, since the Group has international ambitions. AXA thus becomes the corporate name of
Mutuelles Unies-Drouot.
See also
www.sunlifeint.com/generic/company/axagroup/intro.htm which seems to be a copy of the official AXA website. Once they settled on A as the first letter, they probably used a dartboard for the middle letter.