Abbreviation is usually the practice of writing a shortened word to save space on paper, or effort on the part of the writer. For example I might write "instr." as an abbreviation for "instrumental", or "vb." as an abbreviation for "verb".
Some abbreviations do indeed contain apostrophes: "international" becomes "int'l".
It is not usually expected that people would pronounce these words out loud in their abbreviated forms.
Contraction is the practice of leaving out parts of phrases like "it is" and "they are", ending up with "it's" and "they're". This is a way of writing down the shortened phrases as they are spoken. The apostrophes do indeed stand in for missing parts of the words.
However there are some spoken phrases that are sometimes written as a single word, where you might expect to see two apostrophes. "Cannot" becomes "Ca'n't" - but by convention we ignore one apostrophe and write "Can't" - see Can a word be contracted twice (e.g. "I'ven't")?
You are asking to abbreviate the word "little". There isn't a standard way to do that in the sense of "abbreviate" I've defined above.
However, in some accents, the "tt" of "little" becomes a glottal stop, and many writers have chosen to depict that as "li'l". It is a good phonetic representation of the way the word is spoken, where the apostrophe represents the small pause, or the glottal stop as it is said.