Final answer:
Aunt Petunia calms Dudley down by calling him 'Dinky Duddydums,' which reflects her coddling nature towards her son in the Harry Potter series.
Step-by-step explanation:
In the story, Aunt Petunia calls Dudley by the nickname 'Dinkie Duddydums' to calm him down.
The nickname that Aunt Petunia uses to calm Dudley down is 'Dinky Duddydums.' This tender nickname shows the affection Petunia has for her son and is used to soothe him when he is upset or agitated. In the Harry Potter series, where Dudley is a character, this nickname is used multiple times to exemplify the spoiling and overprotective nature of Aunt Petunia towards him.