Simmer 1/4 cup water and lavender in small saucepan until reduced by half, 1 minute.
Add sugar; stir to dissolve. Simmer 2 minutes to thicken syrup slightly. Stir in honey. Set aside.
Bring cream and milk to simmer in heavy large saucepan.
Whisk yolks to blend in large bowl.
Gradually whisk hot cream mixture into yolks. Return mixture to same saucepan.
Stir constantly over medium-low heat until custard thickens and leaves path on back of spoon when finger is drawn across, about 10 minutes (do not boil).
Whisk lavender mixture into custard. Cool mixtures, stirring occasionally, then chill overnight.
Strain and discard lavender.
Process ice cream custard in ice cream maker according to manufacturer's instructions.
Freeze in covered containers. (Can be made 1 week ahead.)