This makes me a little crazy:
The spelling, I mean:
That is just wrong.
There's a historical reason for this spelling, although it isn't addressed by the rules you learned in school. Unlike the apostrophe in contractions such as don't, wouldn't, or he's; or the possessive apostrophe in Abner's (singular) and folks' (plural); or the apostrophe-of-omission in fo'c'sle (forecastle) or bo's'n (for boatswain), the apostrophe in li'l doesn't represent a missing letter—even though three letters are indeed missing.¹ Its function is to create something called eye dialect: nonstandard spelling that approximates a pronunciation that's not much different from the standard, but gives the reader the impression of "dialect, foreign, or uneducated speech" (per Wikipedia). The apostrophe represents "flapping," that thing we do with the t sound in little so that it almost disappears.
This special apostrophe has a wonderful name all its own: the apologetic apostrophe. It shows up in Scots words such as ta'en (taken) to suggest that a letter is missing. It isn't: the word was originally spelled tane. Unapologetically Anglocentric Brits, however, inserted the apostrophe to "prove" that Scots was a degraded form of "pure English."
So where did my Oakland neighbor Dreyer's², which owns the Drumsticks brand, get the apostrophe-at-the-end "Lil'"?
I blame rap music.
Consider Lil' Kim, Lil' 1/2 Dead, Lil' Fizz, Lil' Jon, and Lil' Malik (now known as Mr. Malik). Wikipedia has compiled a much longer list of Lil' rappers.
This is apostrophe abuse, plain and simple.³ What's the function of that terminal apostrophe? Does it represent a missing letter? Is Kim's real name Lily? Or Lilt? Is Mr. 1/2 Dead honoring the French city of Lille?
Or is it mere ornamentation—the orthographic equivalent of a hood ornament?
I doubt very much that it's apologetic. I mean, have you seen Lil' Kim?
Eccentric or nonstandard spellings may be part of the rap brand. But Dreyer's has no excuse for its errant apostrophe. I beg you, Dreyer's, think of the children! You know: the li'l ones.
¹ Yes, the possessive apostrophe represents a missing letter. Long ago, English had a genitive case equivalent to the one in German; "of/belonging to the man" was spelled mannes. As pronunciation changed, the -es contracted into 's.
² Sold as Edy's on the East Coast.
³ For a different strain of apostrophe abuse, see "There's an Apostrophe in There Somewhere," in Arnold Zwicky's blog.