Finally a question where there is a ton of research that I can dig through, I'm sorry if this is a bit long winded. There appear to have been a great deal of different results, heres a few examples:
Weather and arthritis symptoms (1985) concludes that there's no significant change in arthritis despite the patients belief that there was.
In a one month prospective, double blind study of 70 patients (35 with rheumatoid arthritis [RA], 35 with osteoarthritis [OA]), severity of rheumatic symptoms was compared to changes in daily weather conditions using a visual analogue scale. The majority of patients (62%) believed that various aspects of weather aggravated their symptoms. There was no difference in age, sex, diagnosis or perceived symptom severity between weather sensitive and weather insensitive patients. No significant correlation was found between symptoms of any patient group or individual and any of 13 combinations of weather features. These results suggest that contrary to the belief of the majority of patients with RA or OA external weather conditions do not significantly influence the day-to-day symptoms of arthritis.
Effect of weather conditions on rheumatic patients (1990) showed positive results:
In a one month prospective study of 62 rheumatic patients--16 with rheumatoid arthritis (RA), 24 with osteoarthritis (OA), 11 with inflammatory arthritis, 11 with fibromyalgia joint pain--swelling and everyday activity was compared with changes in daily weather conditions. In most patients weather changes increased arthritic symptoms. Women were more sensitive to weather than men (62% v 37%). Pain was affected positively by barometric pressure and temperature in RA, by temperature, rain, and barometric pressure in OA, and by barometric pressure in fibromyalgia. These results support the belief of most rheumatic patients that weather conditions significantly influence their day to day symptoms.
On the belief that arthritis pain is related to the weather (1996) says no it's probably in peoples heads:
We studied patients (n = 18) for more than I year and found no statistically significant associations between their arthritis pain and the weather conditions implicated by each individual.
Rheumatoid arthritis patients show weather sensitivity in daily life, but the relationship is not clinically significant (1998) points out that of all the variables effecting pain, weather is only a small part of that:
In general, patients with higher levels of self-reported pain demonstrated more weather sensitivity. When considering the magnitude of these effects, however, weather variables accounted for only a small amount of change in pain scores. This pattern was true even for patients with the most pronounced pain–weather relationships. Thus, although weather sensitivity was found, the effect sizes were not clinically meaningful.
Which is the dominant factor for perception of rheumatic pain: meteorology or psychology? (2009) examined how strong the belief was when the subjects spent a year in a different weather:
[84%] of subjects belive in an association between weather and rheumatism, while 57% claimed to have ability to forecast weather. The maximum correlation coefficient between weather and arthritis symptoms was −0.451 and the maximum contribution of weather on symptoms was 17.1%. Arthritis symptoms were significantly contributed by Beck depression score. The belief about presence of weather–arthritis association was found to be stronger than its statistical power.
Does the weather really matter? A cohort study of influences of weather and solar conditions on daily variations of joint pain in patients with rheumatoid arthritis (2009) concludes that it depends:
Weather sensitivity seems to be a continuum and a highly individual phenomenon in patients with RA. In the present sample, pain was significantly associated with 3 or more weather variables in 1 out of 6 patients, for whom the magnitude of weather sensitivity might significantly influence pain reporting in clinical care and research.
What's in season for rheumatoid arthritis patients? Seasonal fluctuations in disease activity (2006) is a bigger study with 1665 that concludes positive results across seasons:
We found definite seasonal differences in RA patients, both subjectively and objectively. RA disease activity was higher in spring and lower during fall. Seasonal changes may play an important role in evaluating disease activity of RA patients and should be taken into account when examining these patients.
Weather effects in rheumatoid arthritis: from controversy to consensus. A review. is a meta-study of the research between 1985-2003 that finds that weather is something that has many variables, and a problem with many studies is that the methods are so different:
CONCLUSION: RA variables are positively correlated with the humidity of the microclimate at the patient's skin. High outdoor relative humidity is unfavorable, but has less influence when there are few barriers for water vapor, like clothes, and when air conditioning is used. High temperature is unfavorable since it increases absolute humidity, but beneficial as well, since it reduces the presence of barriers, and stimulates the use of air conditioning. The classic opinion, "Cold and wet is bad, warm and dry is good for RA patients," seems to be true only as far as humidity is concerned.
It seems to me that from the point of view of your father, if he experiences less pain in one weather than another then that should be what he should go on. It appears that what patients experience are so individual that it's hard to pin down an optimal weather condition that can be broadly suggested for everyone. It also seem that weather would be a small portion of pain management, or the research would show a strong correlation between certain conditions and others.
Looking into this reminds me of a quote from Niel deGrasse Tyson about cancer and cell phones (who - like me - is not a Doctor):
If you try to measure a phenomenon that does not exist, the variation in your measurement will occasionally give you a positive signals, as well as negative signals. If your idea is that A causes B, in this case cell phones causes cancer. Then a paper gets written about that result and then people might get concerned that cell phones cause cancer or that the power-lines might cause cancer, this go way back and so if you look at the full spade of these studies even those they don't publish because it was not a positive effect there are some cases where there is in fact less cancer and so these are the phenomenon of a no-result.
When you actually have A causing B the signal is huge and it's repeatable in time and in place in cell phones that repeatable signal is yet to emerge from the total experiments that are done with it. [..]
Maybe what we are seeing is the noise of a no-signal or maybe the jury is still out as they are actively publishing research in it.