Is there a spell or power that has a charge up effect like Kamehameha or Wave Motion Guns?

If not, how would you construct it?

I'm not aware of any options in 5e, but in 3.5e there were spells like Channeled Pyroburst, which scaled based on how many actions were committed:

(Summarized for 5e, damage scales by level) 

Bonus Action — 1d4 Fire damage (Single Target)
Action — 1d6 Fire damage  (10ft radius)
Full-Round — 1d8 Fire Damage (15ft radius)
2 Rounds — 1d10 Fire Damage (20ft radius)

It's a complicated mechanic for 5e, so I wouldn't recommend emulating it this way. Instead I would do something more straightforward:

1) The Player would take the Focus action by stating which spell they want to amplify, then accumulate 1 "Charge Point".

2) The Player can continue to take the Focus action on subsequent turns, to gain up to a number of "Charge Points" equal to their casting modifier.

3) The Player then releases the amplified spell which adds a number of d6's equal to the total Charge Points to one damage roll of the cast spell.

The Charge Die increases as the player levels: 5th (d8), 11th (d10), 17th (d12)

If the player takes any action other than Focus or Cast a Spell (Amplified), the Charge Points immediately vanish and the spell slot is lost.
If the player loses concentration, the Charge Points immediately vanish and the spell slot is lost.

 So, in practice, a 3rd level wizard could spend up to 3 rounds "charging" Scorching Ray  to increase the damage of one ray to 2d6+3d6.

A 20th level wizard could spend up to 5 round "charging" the same spell to increase the damage of one ray to 2d6+5d12.

That's a lot of damage for a single 2nd level spell slot, but 5 rounds of doing nothing is an eternity in combat. 

Magic Missile would need to be explicitly banned from this.

delayed blast fireball

Memnosyne is onto a good system. I would suggest adding the requirement that charging up takes one's concentration--certainly in the case of a spell.