Is the ranged attack resulting from a monk/rogue's Deflect Missiles eligible to add sneak attack damage? For example, if the creature who made the initial ranged attack is engaged in melee as per the rules of Sneak Attack, or if the Monk/Rogue were to somehow have advantage on the roll, for example by way of a spent inspiration point.
I haven't come across anything that seems to openly rule it out, but the idea of throwing somebody's crossbow bolt back at them and doing 1d8 +3d6 damage (at Monk 14/ Rogue 6) seems a little broken to me.
The rules for sneak attack state that:
Most thrown weapons are actually melee weapons with the thrown property, and using something for a ranged attack does not make it into a ranged weapon (official confirmation by JC). So you will be able to sneak attack with a caught dagger (as it has the finesse property) or a dart (as it is a ranged weapon), but not with a handaxe.
As for caught missiles, they are not technically weapons, they are ammunition. This means they count as an improvised weapon. Thus they will not have any properties (ie. finesse). Whether an arrow or bolt counts as an improvised ranged weapon is up to the DM.