Deflect Missile attacks are considered attacks with a monk weapon according to the feature description's last sentence and the Sage Advice compendium question relating to it.
As such, due to the Martial Arts class feature, you are allowed to choose to use STR or DEX for the attack and damage bonuses. You also get your proficiency bonus for the attack roll as specifically stated by the Deflect Missiles class feature. In addition, the usual Martial Arts monk weapon rules of being able to choose the weapon's normal damage dice (the projectile/weapon's damage) or Martial Arts' monk weapon damage dice applies.
When a monk using Deflect Missiles catches and throws a projectile, what is the damage of the attack?
A missile counts as a monk weapon if thrown via Deflect
Missiles; it deals its damage or Martial Arts damage (the
To summarize, the attack roll is d20+(DEX or STR)+Proficiency, and the damage roll is (original weapon damage dice or Martial Arts' monk weapon damage dice) + (DEX or STR).
Worth noting is that, unlike the original ranged weapon attack targeted at the monk, you are limited to a 20/60 range on the Deflect Missiles attack as clarified by the official PHB errata.
With the errata, the end of the latest Deflect Missiles feature description now contains:
You make this attack with proficiency, regardless of your weapon proficiencies, and the missile counts as a monk weapon for the attack, which has a normal range of 20 feet and a long range of 60 feet.