Paul Krugman has had a few posts on this question, most recently this one, the first one here. Krugman is right in asserting a major role for air conditioning, but there is a subtle framing point which is sometimes neglected. The most on-point study is this piece from Jordan Rappaport (pdf):
U.S. residents have been moving en masse to places with nice weather. Well known is the migration towards places with warm winters, which is often attributed to the introduction of air conditioning. But people have also been moving to places with cooler, less-humid summers, which is the opposite of what is expected from the introduction of air conditioning. Nor can the movement to nice weather be primarily explained by shifting industrial composition or by elderly migration. Instead, a large portion of weather-related moves appear to be the result of an increased valuation of nice weather as a consumption amenity, probably due to broad-based rising per capita income.
Overall Rappaport concludes that “nice [warm] weather is a normal good” is the more important driving force behind the movement to the Sun Belt than is air conditioning per se, though of course air conditioning makes nice warm weather all the nicer. Evidence from compensating differentials also indicates that “…the decreased discomfort from heat and humidity afforded by air-conditioning has not been the primary driver of the move to nice weather.” (p.26)
From 1880 to 1910, Americans overall are moving to places with bad (cold) weather. In the 1920s they start moving, on net, to places with nicer weather and that trend has not let up. The arrival of affordable air conditioning in the postwar era bumps this up a bit, but the main trend already was in place. Furthermore air conditioning has been in the south for quite a while now, but migration in that direction continues. In his second post on the topic, Krugman refers to this as a “gradual adjustment” to AC, but it seems to better fit the nice weather as a normal good story. We’ll know more if we see this migration continuing, but I expect it will. At some point it won’t be plausible to call the ongoing movement a “lagged response” to the introduction of air conditioning, but again it will fit the normal good story pretty smoothly.
Note also that life expectancy is notably higher in warm weather than cold weather. Deschenes and Moretti conclude (pdf): “…The longevity gains associated with mobility from the Northeast to the Southwest account for 4% to 7% of the total gains in life expectancy experienced by the U.S. population over the past thirty years.”
That again points toward a “normal good” explanation, with air conditioning playing a supporting role.
That all said, if you look at the larger political debate going on here, Krugman is correct in arguing that lower taxes are the not main reason for this migration, even though the median voter in these states probably approves of such relatively low tax rates. In any case, there is a clearer and better version of the weather hypothesis which can be put forward.