I gave her a name; it was ‘Betty’.

The reason the period goes inside the quotation marks has to do with lead type. That’s lead, as in the metal. Print used to be set as individual letters, each letter on its own piece lead. The period is a very narrow letter. A double quotation mark is a bit wider. Printers used to lose the periods off the end of sentences. Whenever possible, they put another bigger lead after a period, especially at the end of a line. Quotations were often on their own lines, thus the quotation marks were often put to the duty of holding the sentence intact.

Periods and commas always go inside quotation marks, even inside single quotes.

Examples The sign changed from “Walk,” to “Don’t Walk,” to “Walk” again within 30 seconds.

She said, “Hurry up.”

She said, “He said, ‘Hurry up.'”