- Place item you wish to alter into the Mystic's Enchant interface and a list of properties that you can reroll (replace) on that item will appear. Click the question mark next to both to see what properties they could reroll into.
- Once you have decided which property to reroll place the materials you need to give the Mystic to do so into the interface and hit Replace Property.
- Select the replacement property from the list of available choices.
Scope and Restrictions
- The Mystic will also offer the original property you replaced so if you don't like the new one you got you can for the price of materials and gold reroll back to the old one.
- Only one property can be rerolled but it can be rerolled repeatedly, each time costing materials and gold to do so.
- Enchanted items become bound to your account (BoA).
- It has not yet been decided if permanent properties on legendary items can be enchanted or just the random properties: 
Development of Enchanting
The first mention of "enchanting" came from the Caravan FAQ, released in August 2010 at the Gamescom expo. There was just a single line about the Mystic, which credited her with Enchanting, but didn't give any indication what that meant.
A quote from an interview with indiablo.com from Gamescom 2010 confirms that the bonuses are not random:
More information on Enchanting was revealed by Jay Wilson in an interview at Blizzcon 2010. 
From the way Jay describes "finding enchantments" it makes them sound like recipes. Recipes are what the Artisans use to craft new, semi-random items in Diablo III's form of gambling. Artisans know some recipes to start with, and acquire more through their level up training. But the best recipes in the game are found as drops from monsters. These recipe scrolls are then given to the appropriate Artisan, and from then on that Artisan can craft that item.
When asked what happens to her enchants Blizzard confirmed:
The Mystic's return was revealed at Gamescom 2013 although details of all her abilities were not revealed the fact she would be able to reroll an affix (enchant) was and welcomed by the majority of the community.