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Oh, puns, we love you so. The phrase "happy medium" long precedes the Medium's appearance as a character in the book: in the first chapter, Mrs. Murry says to Meg, "A happy medium is something I wonder if you'll ever learn" (1.63). Later the twins echo her words, telling Meg to "use a happy medium, for heaven's sake" (2.16). The hint is that Meg swings wildly from one extreme to another, and if she could just cool down and chill out she'd be much happier.
So what does this happy medium have to do with the Happy Medium? That's where the pun comes in: the Happy Medium lives on a planet where everything has been evened out to a medium gray, but she herself is a different kind of "medium," the kind that works for the Psychic Friends Network. And she is indeed happy, though it seems she gets to be that way through pretending unhappy things don't exist: as she says to Mrs. Whatsit, "Oh, why must you make me look at unpleasant things when there are so many delightful ones to see?" (5.84). Her cheerfulness is not that of a person who looks at the world straight-on and still finds cause to be optimistic, but of some who maintains a positive mood by sticking her fingers in her ears and singing "la la la" when anything upsetting comes up.
The Happy Medium's power to show the kids the shadow of the Black Thing over Earth strengthens their will to fight. She also offers another kind of lesson, to Meg at least, that maintaining a happy medium, or a Happy Medium, at the cost of self-enforced ignorance is not a worthwhile goal.