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A short way to explain is by the following poem:
In search from A to Z they passed,
And "Marguerita" chose at last;
But thought it sound far more sweet
To call the baby "Marguerite."
When grandma saw the little pet,
She called her "darling Margaret."
Next uncle Jack and cousin Aggie
Sent cup and spoon to "little Maggie."
And grandpapa the right must beg
To call the lassie "bonnie Meg."
From "Marguerita" down to "Meg,"
And now she's simply "little Peg."
I also found this .......
A: “Margaret” has spawned a bewildering variety of nicknames, from “Margot” to “Daisy.”
The name itself is a variant of the obsolete word “margarite,” meaning “pearl” or “precious stone.” It is apparently of ancient Asian origin, filtered through Greek, Latin, Teutonic and Old French.
It became a very popular given name in medieval England and Scotland, where it was conventionally taken to mean “pearl.”
Since “Margaret” is quite a mouthful, nicknames soon spun off from it. Perhaps the strangest is “Daisy.” It’s a pun dating to a time when “margaret” was also an English slang term for the ox-eye daisy. It became an independent first name during the 1900s fad for flower-based names.
More common were such shortenings and diminutions as “Maggie,” “Meggie” and “Meg.” Some etymologists say that the diminutive “Maggie” form came first, with “Meg” following as a shortened form; others say the shortened form “Meg” came first, with “Maggie” and “Meggie” following as diminutive forms. Nobody really knows.
What we do know is that “Peg” is an altered form of “Meg,” and “Peggy” is an altered form of “Meggie.”
“Meggie” and “Meg” were distinctively Scottish nicknames, so “Peg” and “Peggy” probably were, too.
But no one knows why the “M” was changed to a “P.” (A theory about Celtic-language influence has proven fruitless.) They are similar consonant sounds; maybe the change was inspired by a nonsense-rhyme nickname like “Meggy-Peggy.”
Interestingly, the same morphology can be seen in nicknames for “Mary.” “Molly” is a pet form of “Mary”; “Polly” is a variant form of “Molly,” with the “M” also mysteriously changed to a “P.”
It’s unusual for a nickname to become an independent first name by itself. But “Peg” and “Peggy” have done so (apparently first occurring in the U.S.), no doubt aided by their lack of clear connection to “Margaret.”