[RPG] Does the Evocation wizard’s Empowered Evocation feature increase the damage of the Booming Blade cantrip

I don't think this has been answered simply.

Jeremy Crawford has said the following about the Magic Missile spell:

It's one damage roll, just like fireball, but that roll can damage the same target more than once.

The specific rule this is taken from is (PHB p.196):

If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them.

And the Magic Missile spell (PHB p.257) states that the missiles strike more than one target at the same time:

The darts all strike simultaneously

Thus the process is:

1. Assign a target for each magic missile created by the spell, you may target one creature more than once
2. Roll damage for the spell of 1d4+1 + [bonus]
3. Apply this damage to the target(s) once for each missile created
4. The missiles hit simultaneously and are each a separate source of damage for damage resistance or concentration rolls etc.

Thus any spell damage bonus from an ability such as Empowered Evocation (PHB p.177) applies once to the damage roll, but that damage roll is applied for each missile.

Potent Cantrip makes a cantrip exactly the kind of effect that Evasion works against. Evasion causes "an effect that allows you to make a Dexterity saving throw to take only half damage" to instead deal no damage. Potent Cantrip means that your cantrips deal half damage on a successful saving throw. Which is to say, the target gets to make a saving throw to take only half damage.