There are plenty of different variables as to why certain people suffer from wine headaches.
Here are some potential culprits and some possible solutions
Histamines - There are a few different histamines in wines that can effect individuals whom have a particular enzyme deficiency that fails to counter the histamines and alcohol, which is much higher in red wine than white. Can be combatted by drinking black tea before and during alcohol consumption
Tannins - evident in skin,seeds and stems of grapes as well as barrels. higher tannin wines are generally from colder climates and more evident in young wines of particular varietals (except beaujolais) and factors of production including pressing down of grapes, full stem maceration. Tannins are necessary to give structure and backbone to wines but these fall away as wines age and therefore drinking old aged wines may be preferred to those sensitive to tannins
The pounding headache some people experience when drinking red wines may be due to individuals whom are not able to metabolize prostaglandin receptors in muscle tissue in the brain. An option is to take aspirin prior to drinking and keeping hydrated
Some individuals are allergic to certain yeast and bacteria in wine
A small % of individuals are allergic to sulphites and will experience anophylatic shock . A note should be made that sulphites are a natural by-product of all wine making and is necessary to preserve product n the bottle. There are also more sulphites in dried fruit and soft drinks than in wine. A well made wine with minimal intervention and low residual sugar should have lower sulphites to combat and free sugars and yeast interactions. (sulphites are generally higher in white wines than red wines)
Look for low tannic, well made, organic, and potentially aged red wines to avoid some of the factors above. Red wines made from thin skin grape, minimal extraction and seed,stem contact will help. Look into trying some good quality Beaujolais, European/ Rsa/Chilean Pinot Noir.