The reason why she calls Pistol "Peesel" is because Mistress Quickly has the curious habit of using the wrong words, or as the eNotes study guide says, "speaking in malapropisms":
Her generosity, her nervous confusion, and her tendency to speak in malapropisms (she uses the word "honeysuckle" for "homicidal" in II.i.50, and "confirmities" for "infirmities" in II.iv.58) frequently leave her vulnerable to Falstaff's swindles and jokes.
One theater critic calls Mistress Quickly the doyenne (or senior member) of malapropism.
Mistress Quickly may well have been mistress of the malapropism, as someone has said, but she was also a dirty-minded little so-and-so and would latch quickly on to any double entendre. 'Peesel' - sometimes 'pizzle' - was a euphemism for 'penis' and its similarity to the Captain's name was too much of a temptation to resist.