Answer:
The right way to write this excerpt would be:
When my little-known cousin Tammy -aunt Edna's second daughter- said "That depends on what you mean by 'on time'", I laughed for at least fifteen minutes. 
Step-by-step explanation:
The errors I can see in this excerpt are:
- The word cousin goes without capital letter.
 - Aunt, again without capital letter.
 - The comma after the word said, and before the quote is wrong.