No, they can't, because mirror image changes the target of the attack to an illusory duplicate.
The Way of the Drunken Master monk gets the Redirect Attack ability as part of their 6th level Tipsy Sway feature (XGtE, p. 34). The description says:
Redirect Attack. When a creature misses you with a melee attack roll, you can spend 1 ki point as a reaction to cause that attack to hit one creature of your choice, other than the attacker, that you can see within 5 feet of you.
The mirror image spell description states:
Three illusory duplicates of yourself appear in your space. [...]
Each time a creature targets you with an attack during the spell's duration, roll a d20 to determine whether the attack instead targets one of your duplicates.
If you have three duplicates, you must roll a 6 or higher to change the attack's target to a duplicate. With two duplicates, you must roll an 8 or higher. With one duplicate, you must roll an 11 or higher.
A duplicate's AC equals 10 + your Dexterity modifier. If an attack hits a duplicate, the duplicate is destroyed.
The key is that mirror image does not simply make the attack miss you; it causes the attack to target one of the illusory duplicates instead. However, Redirect Attack takes effect when "a creature misses you with a melee attack roll". If mirror image changes the target of the attack entirely, the attack roll neither misses nor hits you - it misses or hits the illusory duplicate (which has its own AC equal to 10 + your Dexterity modifier).
Because the effect of mirror image is to change the target of the attack from you to an illusory duplicate, features triggered by an attack missing or hitting you will not be activated when mirror image changes the target to an illusory duplicate.