Honey syrup is a syrup made from honey and water -- often used as a sweeter, fuller-bodied alternative to simple syrup, you make it the same way (half water, half honey).
Ask yourself: how will this honey balance with the other flavors in the cocktail? Are the spirits and other ingredients you're using light and herbal? Bold and aggressive? Somewhere in between? Intense buckwheat and chestnut honeys wouldn't work well in, say, the Bee's Knees, which is made with gin and lemon juice. But clover honey will work well, and orange-blossom honey will be even better.