Hangovers are generally caused by dehydration as a result of consuming larger amounts of alcohol and/or as a reaction to additives in alcoholic beverages. Because organic drinks normally are of purer quality and avoids many of the additives one often find in non-organic products, it is possible that hangover symptoms could be reduced.
Some people claim organic wine, drunk in moderation, is less likely to give you a hangover than standard wine. All wines are treated with sulphur dioxide to help prevent deterioration in taste, but most organic winemakers are only allowed to add far less than is routine practice. Many people react strongly to sulphur and get headaches from it, so it could be fair to say that organic wine is less likely than normal wine to give you a hangover.
This being said, the only way to prevent a hangover is to avoid heavy drinking and limit the amount of alcohol you consume to an amount that your body can metabolize.