Happy words! Because…well, joy is mentioned a lot this time of year, I guess. So here we go.
Happy first showed up in the late fourteenth century, where it meant lucky—maybe it changed because if you’re lucky, you’re probably happy. It’s actually a mix of the word hap (more on that in a second) and the suffix -y, which actually means “full of or characterized by” like hearty or funny. So this means that happy means full of hap. But what’s hap? I didn’t even know it was a word! But it is, having shown up in the early thirteenth century meaning chance, fortune, or fate. It comes from the Old Norse happ, good luck, Proto Germanic hap, and Proto Indo European kob-, to suit/fit or succeed. Hap basically means luck (generally good), but it also has a verb version that means to come to pass. Which is what gave us happen. And happenstance. And perhaps.
Joy showed up in the early thirteenth century, coming from the Old French joie, pleasure, and classical Latin gaudia and its singular gaudium, which both just mean joy or pleasure. They come from the verb guadere, rejoice, which can be traced even further back to the Proto Indo European gau, also rejoice. That also might be where gaudy comes from, by the way.
Glad comes from the Old English glaed, which could mean happy, bright,or brilliant (those last two definitions are going seem a lot more significant in a few seconds). Glaed comes from the Proto Germanic glada-, which is from the Proto Indo European ghel-, shine. The origin word for glass, gold, and yellow. So because things are figuratively bright when you’re glad, it comes from the word for shine.
Cheer first showed up in the early thirteenth century meaning, get this, the face, as in expressing emotion. It comes from the Anglo French chere, the face, and Old French chiere, face or expression. Before that it was cara in Late Latin (also face), which probably comes from the Greek kara, which means head or skull. That’s actually from the Proto Indo European root ker-, head or horn—it’s literally the origin word for horn. So because your face is on your head and your expressions are on your face…you get cheer? Is that how it works?
tl;dr: Happy words are weird.
Tony Jebson’s page on the Origins of Old English