A quick question.  A creature moved out of melee range causing an opportunity attack.  Our rogue used Booming Blade with his short sword for the opportunity attack.  The question, can the rogue use Booming Blade with this opportunity attack?

No, they can't, unless they have some special feature that allows them to cast a spell in place of an opportunity attack. The most common example of this is the War Caster feat.

In the absence of a feature like that, an opportunity attack allows you to make one melee attack or unarmed strike. It doesn't allow you to cast a spell (even one like Booming Blade that involves making an attack).

The SAC has this answer. It's using True Strike for the explanation, but IMO the cantrip is similar to Booming Blade (or Green-Flame Blade and Vengeful Blade).

That "special feature" might be War Caster.

Can you use True Strike with Extra Attack, Opportunity Attack, Sneak Attack, and other weapon attack options?

True Strike doesn’t work with Extra Attack or any other feature that requires the Attack action. Like other spells with a casting time of an action, casting True Strike requires you to take the Magic action, not the Attack action. Similarly, unless a special feature allows you to do so, you can’t cast True Strike when making an Opportunity Attack.

However, an attack made as part of True Strike works with Sneak Attack so long as it fills the normal requirements for that feature. For example, if you have the Sneak Attack feature and cast True Strike with a Finesse weapon, you can deal Sneak Attack damage to the target of the attack if you have Advantage on the attack roll and hit.

EDIT: Since the question is about Booming Blade , this is the 2014 SAC answer that mentions it: Can you use green-flame blade and booming blade with Extra Attack, opportunity attacks, Sneak Attack, and other weapon attack options?