Arcane Shot Options can only be used while you are firing an arrow
The Arcane Shot feature states:
when you fire an arrow from a shortbow or longbow as part of the Attack action, you can apply one of your Arcane Shot options to that arrow. You decide to use the option when the arrow hits
We can interpret this either as "triggers" or as "conditions". When I say "trigger" I mean a singular event that allows a subsequent option. "Conditions" as I am using the word refers to a state of affairs during which an option can be used.
Using the trigger interpretation there are two events described in the text. The first event is "when you fire an arrow from a shortbow or longbow as part of the Attack action" while the second event is "when the arrow hits". If we consider these as isolated events then you could use an arcane shot option with Deflect Missiles. You have fired an arrow as part of the attack action and when the monk makes her attack the arrow does hit.
The problem with this interpretation is that it doesn't define when the first trigger ends except when the arrow hits. Without defining an end every arrow you fire would qualify for the second trigger in perpetuity. The simplest exploit with this is to get around the "as part of the Attack Action" clause. The Arcane Archer spends a couple of turns firing arrows into the air. She then collects the arrows and later, when she readies an attack, she uses them as a reaction. When they hit they satisfy the second trigger and so she uses an arcane shot option with them.
This is, however, obviously not how the feature was intended to function. It is ridiculous to imagine that any arrow that has ever been fired as a part of your attack action can trigger Arcane Shot at a later time. Here is where the "conditions" interpretation becomes relevant.
Instead of viewing "when you fire an arrow" as an instantaneous event we can think of it as a process. "When" and "While" can sometimes be used synonymously so if we interpret the sentence as "while firing an arrow" we understand that the condition must continually be satisfied for its consequences to activate.
Using this interpretation it is clear that the deflected arrow could not benefit from Arcane Shot. Once the arrow hits the monk you are no longer "firing an arrow as part of the Attack action" The fact that you previously fired that arrow is as irrelevant to the monk's attack as it would be to your readied action if you saved arrows for later.