Learn English – “Pretty please with sugar on top”

Etymonline says

For Pete's sake is attested from 1924, probably a euphemistic use of the disciple's name in place of Christ;

W.S. Farmer & J.L. Henley, Slang and Its Analogues, vol. 2 (1891), says that "dead to rights" means "certain; without doubt," and asserts that it is simply an amplification of the earlier term "to rights," meaning "completely to one's satisfaction." Dead appears in a similarly amplifying way in such current phrases as "dead broke," "dead certainty," "dead heat," and "dead ringer."

Christine Ammer, The American Heritage Dictionary of Idioms (1997) gives a sense of how much the meaning of the phrase has evolved:

dead to rights In the act of committing an error or crime, red-handed. For example, They caught the burglars dead to rights with the Oriental rugs. This phrase uses to rights in the sense of "at once."

Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960) gives both meanings:

dead to rights 1 Certain. c1880. Colloq. 1939: "You've got him dead to rights...' Gardner, D. A. Draws, 185. 2 Caught in the act of or irrefutably accused of an illegal, immoral, unethical, or antisocial act.

But the Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) agrees with Ammer's narrower definition:

dead (or bang) to rights adv phr by 1859 With no possibility of escape or evasion; in flagrante delicto; red-handed: ...was caught "dead to rights" and now languishes in the city Bastille—San Francisco City Argus

UPDATE (9/17/2013): I did some checking in Google Books search results and found a source that probably explains Chapman & Kipfer's "by 1859" dating. George Matsell, Vocabulum; or, The Rogue's Lexicon (1859), contains this entry:

DEAD TO RIGHTS. Positively guilty, and no way of getting clear.

To like effect is a glossary entry in George Burnham, Memoirs of the United States Secret Service (1872):

DEAD TO RIGHTS, caught, with positive proof of guilt.

So it appears that Farmer & Henley didn't fully grasp the sense of "dead to rights" as the phrase was used in the latter half of the nineteenth century, though the meaning that Slang and Its Analogues ascribes to the term certainly does appear in some contemporaneous writings. For example, from "Born and Raised in Mobile," in Sweet & Knox, Three Dozen Good Stories from Texas Siftings (1887):

"You never heard what became of Wes Brutus?" inquired the driver.

"No; I expect he's sloshing around, though. Wes was a bully boy."

"Well, now, pardner, your [sic] just dead to rights; he was for a fact—ah! you folks are going to try Foot & Walker's line, are you? That's right; that's correct.

FURTHER UPDATE (1/24/2017): The earliest match for "dead to rights" that an Elephind search of multiple old newspaper databases returns is from "Pickpockets at the Depot: Caught in the Act—One of the Gang in Prison," in the [Harrisburg] Pennsylvania Daily Telegraph (August 14, 1862):

This morning, upon the arrival of one of the trains on the Northern Central Railroad, a passenger named Loring Gillis, of Buffalo, New York, felt a hand inserted in one of his pockets, and before it could be removed, he turned suddenly around, and succeeded in coloring [sic] its owner, a dapper, flashy dressed individual of thirty or thereabouts, with "pickpocket" written all over him. Upon feeling for his pocket-book, containing about two hundred dollars, Mr. Gillis found it was gone. The "chevalier," like all his class when caught "dead to rights," tried to shut Mr. Gillis' eye up with the "virtuous indignation" game, but finding that no "go," tried to close it with one of his fists. Here, too, it was foiled, for Mr. Gillis showed a superior proficiency in the "manly art," and the "chevalier" was finally forced to succumb. During the scuffle two individuals, supposed to be pals of the pickpocket, were seen to run from the crowd, but from some cause were not pursued and arrested.