This post is the result of that brainstorming, of course. Please note that I decided to focus on games that have penguin protagonists. As such, you won't see games like the aforementioned platformer or, say, Super Mario 64 discussed here.
1. Binary Land (Famicom)--This Hudson Soft-made game, which puts players in control of a pair of in-love penguins, is one of my favorite Famicom games, so it shouldn't be too surprising that I'm including it here. I don't love Binary Land because of its adorable protagonists, by the way; I love it because of its unique gameplay (you literally control both characters at the same time as you work them toward each stage's "exit" of sorts) and its highly hummable soundtrack.
2. Parodius Da! (PC Engine)--I could have included any of Konami's Parodius games in this post, really, but I decided to go with this, the series' second release, because it's the first I played. A few additional reasons for its inclusion here: it stars Pentarou, son of Antarctic Adventure's main character, and it features hordes of penguin "baddies" as well as a rather notable penguin boss (who earns extra points from yours truly for being both pink and a pirate), too.