Talonsoft for Beginners!
If I was trying to get someone hooked on computer wargaming, I would start with Talonsoft's Gettysburg (Battleground Series). It's a bit dated, but:
1. Extremely popular topic (can combine exposure to the game with a viewing of the Gettysburg movie).
2. Decent graphics and sound (even though you eventually want to turn off the videos)which makes it visually appealing.
3. Historically accurate so you can read about the battle and then relive the action.
4. Game mechanics are fairly simple and easily taught.
Games like COAW are great, but really a bit overwhelming for someone getting their first exposure to wargaming.
Talonsoft's Campaign Series would be my next choice if someone really likes the idea of modern warfare and small unit action.