How long must a property be held before it can be exchanged?

There is no minimum holding time for an original or replacement property before it can be exchanged for another property. 

Timing of sales of investment properties must be made on the basis of when you think you can make the most profit.  If someone offers you double your money a month after you acquire a property, you would be crazy to turn that down. 
 
However, it doesn't look good to flip too many properties too quickly.  1031 exchanges are only available to investors and not to Dealers.  Buying & reselling a lot of properties in a short period of time is one of the many elements in deciding who is a Dealer.  The longer a property is held, the less it looks like dealer inventory and more like an investment asset.