In a lot of my saves, I'm just really basic and have names like "Karaoke Club" or "Book Club," LOL.
For my current save, I'm being a little more creative. The society is similar to 1800s England, so I have upper, middle, and lower class sims. Some of the club names are based on actual clubs at the time. Here are some clubs I'm forming:
Rose & Crown - upper class males; reading and drinking bar drinks (they'll have their own very fancy club house, with staff)
Almack's Coterie - upper class and a few notable middle class; dancing, eating, drinking
Rafe's Club - upper and middle class males; fighting, playing cards, and drinking bar drinks
Refined Ladies' Private Tea Club - the wealthier middle class females; socializing, drinking tea (and generally being snobby toward the less well off middle class women)
I also have clubs that are designed for unmarried sims to meet each other in their own social class. There will be dancing and eating. The upper class will meet in a beautiful building that's specifically for balls, and the middle and lower classes will meet in a large room at the church:
Upper Class: Ladies' Cotillion, Gentlemen's Cotillion
Middle Class: Women's Joy Club, Men's Joy Club
Lower Class: Women's Hope Club, Men's Hope Club