A month (time between consecutive new moons)is about 29 and a half
days, on average.  How great is the variability?  i.e., roughly, what
are the longest and shortest times that it can be over the year?

I understand your definition of a month to be the cycle between lunar
phases.  If that is correct, 29.27 to 29.83 is about how many days are
in a month.

Based on the information contained at
http://en.wikipedia.org/wiki/Tropical_month#Synodic_month

Good.  I should also have asked, but didn't, how it correlates with
time of year, presumably the main variable.

Anyway, thanks.

The variability of lunar month can be from about 28 days to 29 3/4
days with average being about 29 1/2 (29.53).  Lunar year is
approximately of 354 days.

The lunar cycle variability is a function of the following lunar orbital elements:

1.  The inclination of the mean equator of the Moon to the true
equator of the Earth.

2.  The arc of the mean equator of the Moon from it's ascending node
on the true equator of the Earth to its ascending node on the ecliptic
of the date

3. the arc of the true equator of the Earth from the true equinox of
date to the ascending node of the mean equator of the Moon.