It was common knowledge in my youth that 7-up got its name from the size of its bottle. Coke and other soft drinks were sold in 6 1/2 oz bottles. 7-up, to distinguish itself, was sold in 7 oz bottles. Get it? 7-up = 7-even.
Link to the column in question.
Common knowledge is, as usual, totally wrong. 
And, not to step on the Master’s toes, but the Mikkelsons really do a better job of explaining that no one really knows why the stuff was named that. Cecil’s answer and the OP’s are merely two among many, and there is no official one.
Does it still have lithium?
Why… yes. Why… do you… ask?
“Coca-Cola hits the spot/8 full ounces, that’s a lot!”
That was Coke’s jingle for decades. Where do you get the 6 1/2 oz thing?
Probably because of the jingle that went: “Pepsi-Cola hits the spot / Twelve full ounces, that’s a lot / Twice as much for a nickel, too / Pepsi-Cola is the drink for you.”
Personal experience. That was the size of the bottles of coke we had delivered to our house by the case in the early to mid 60’s. Always thought the size was strange. I grew up in St. Louis, where incidentally, 7-up was originally bottled. Have no particular memory of the 7oz bottle, though.
Whoops. 
The website BOTTLE HISTORY confirms the 6 1/2oz Coke bottle from the mid-50s all the way through 1986. The pdf Society for Historical Archaeology - Archaeology of the Modern World confirms the 7oz 7-Up bottle in the same period.
Note that the name dates to 1929, so any explanation depending on facts from decades later is right out.
And the lithium was removed decades ago.
Many years ago I read something talking about soda origins and the “up” part of the name was due to the Lithium. The article claimed the Up part was a known entity, 7-up, due to the Lithium, being a drink that would perk you up, and that the 7 was believed to be due to the 7 ingredients, though in the article as in Cecil’s and the Snopes articles it mentioned that it was unknown if this is where the 7 actually came from, as the inventor had never answered that question.
No idea where I saw this article as it was pre-internet when I read it, so take that for what you will. I do know from my grandmother that 7-up was considered a lady’s drink by her social circle in the era during and shortly after prohibition, and the gentlemen would drink alcohol or coke but that a lady drank neither, the lithium was a much more socially acceptable medicine for a lady. Not sure how widespread that attitude was.
One fact ignored by all of these explanations is that Grigg did not make up the name “Seven Up.” It was the existing name of a card game, which dates back to at least 1830, according to the OED. Grigg, for whatever reason (and ascertaining that reason is the point of this exercise), took that established name and applied it to his lemon-lime soda.
Another fact that may be relevant: According to Snopes and Wikipedia, 7Up was invented in October 1929. However, the U.S. Patent and Trademark Office record shows that “Seven Up,” the trademark, was first used in commerce on October 4, 1928. This implies that the name was devised first and the beverage later, which would lead us to discount explanations that are specific to the beverage (e.g., that it contained seven ingredients).
Of course, this line of reasoning is dependent on the dating on Snopes and Wikipedia, so I’m not sure how far to push it. The dating in the trademark record is probably pretty solid.
The trademark office shows that the submitter, Griggs, says that it was first used in commerce Oct. 4, 1928. I’m sure Griggs meant that the beverage was called that at that date. The trademark office publishes this, allows time for people to object, and then grants the trademark. That was Oct. 1929.
Use in commerce was then a strict legal requirement. Typically the use was fairly minimal, just enough to show that the trademark had actually appeared on products or advertising. This was not something where Griggs would have been willing to take the risk of perjury, especially as there was no apparent reason to do so.
I made a mistake on my earlier post. The first use in commerce was August 7, 1928. Howdy Corporation (Griggs’s company) then applied for a trademark on Oct. 4, 1928. This application was posted in the Official Gazette on Nov. 13, 1928 (for this date I’m relying on the OED, not USPTO records). Registration was granted on Feb. 5, 1929.
I think the beverage was already invented before the application for trademark.
Sounds plausible, and that’s the way things normally work, but the (admittedly weak) evidence presented to date implies that the mark came first, presumably being applied to some other, related beverage product. It would take very little to convince me that the beverage did indeed come first, but so far we don’t even have that.
I once heard / read that it possible has something to do with the atomic weight of Lithium (which is 7), in any case it’s a nice coincidence!
The periodic table was reported around 1870, so I guess on the time-scale likeliness it is a possibility.
Like a lot of brands, 7Up was used because it sounds nice. Seven has always been a lucky number, and up is a positive attribute. (Things are looking up!). Up also connotes lightness which is a good thing to have in what is suppose to be a light, refreshing drink.
7up is short, so it’s easy to paint a nice big ad in a small area, and it’s easy to remember. Doesn’t do much good if your target audience can’t remember your brand name (“I’ll have a bottle of the Bi-Label Lemonated Lemonade …uh… Never mind, just give me a Coke.”)
That’s probably all there is to it. It just sounded nice. Think of Heinz 57 varieties.
Save those braincells for contemplating more important tasks. (How do they get those seven tomatoes into those tiny cans of Hunts Tomato paste?)