I second both "Space Seed" and "The Trouble With Tribbles".
"Space Seed" because it introduces Khan and you'd have to have lived under a rock for the last 35 years not to know who Khan Noonien Singh is.
"The Trouble With Tribbles" because it is one of the most famous episodes outside Trek-dom. Bonus points if you watch "Trials & Tribble-ations" from DS9, too. In fact, I'd probably watch that episode of DS9 as a "gateway episode" to TOS.
If he's a gamer, get him hooked on STO (Star Trek Online). Then he'll have to work his way through all the series so that he can keep up with the 'verse.