The bar is younger than its name by more than two centuries, and nobody can prove where Rowntree's found it.
The strongest lead is the Kit-Cat Club, a Whig political club whose members backed a Protestant succession. Its dates and meeting places are unsettled. Culinary Lore says it formed in 1703 and met at a tavern on King's Street, Westminster. Wikipedia's article on the club says its first meetings were at a pie-house in Shire Lane, and that its origins may reach back into the 1690s.
The club's name is most often traced to the pie-house keeper, Christopher Cat. The surname is spelled Cat in the Wikipedia KitKat and Daily Meal accounts, Katt in Culinary Lore, and Catt, Cat, Katt or Catling in Wikipedia's club article. The food is also disputed. Wikipedia says mutton pies called kit-kats were served at meetings, while Culinary Lore says tarts.
So the club story has loose ends, and the link from the club to the chocolate is only inferred. Wikipedia notes only that the food name goes back to the club. It does not say that is where the bar's name came from.
What the records do show is a trademark. Joseph Rowntree's firm in York registered both "Kit Cat" and "Kit Kat" in 1911, according to Culinary Lore and Wikipedia. It did not use them at once. In the 1920s the name appeared on a boxed line of chocolates, which was later dropped.
The wafer bar came next. It launched in 1935 as Rowntree's Chocolate Crisp. Culinary Lore says the marketing director, George Harris, renamed it in 1937. It gives the new name as Kit Kat Krisp, while Wikipedia says Kit Kat Chocolate Crisp.
Other explanations circulate, and the sources do not back them. Culinary Lore rejects a popular story about a West End nightclub, because the wafer did not exist when the name was registered. The Daily Meal mentions a sorority slogan story that it says lacks evidence. It also notes that "kit-kat" was a fashionable word at the time, with a magazine of that name. Culinary Lore says that was a Philadelphia weekly in the late 1800s.
The honest summary is that the name belongs to a well-documented 1911 trademark and a poorly documented club from around 1700. The advertising slogan, "Have a Break, Have a Kit Kat," came much later, in the late 1950s, according to Wikipedia.