This claim is somewhat true based on the facts at hand. I believe it is true, but make your own decision based on the facts.
A study published in 1997 in the International Journal of Biometeorology has a few facts present which are relevant here.
1) "Mean pain and rigidity scores for each time of each day were found to be correlated with the meteorological data. Correlations between mean symptoms and temperature and relative humidity were significant (P < 0.001)"
2)"Stepwise multiple regression analysis indicated that meteorological variables and time of day accounted for 38% of the variance in mean pain and 20% of the variance in mean rigidity when data of all months were considered."
This means you can be 99% sure that the pain and rigidity felt by arthritis patients was not explained by nothing, and was instead explained by variables of temperature, relative humidity, barometric pressure, wind speed and precipitation.
In addition, the changing weather variables explained 38% percent of the variation in pain. The significance of this number is debated. Some researchers say only 25% percent of the variation should be explained by regression to be substantial, whereas others say it should be above 50% or even up to 75%. However, they would all agree that 38% is too much to be insignificant.
So, the 1997 study was able to show that weather variables like temperature and pressure had a real effect on the pain felt by arthritis patients. Thus, someone with arthritis could possibly predict a change in weather based on the amount of pain they felt that day, especially someone who has no other factors affecting their arthritis pain. If this person's arthritis suddenly flared up, and they lived in an area where pressure affected weather, they would be able to predict a change in weather.