The Facts on File Dictionary of Word and Phrase Origins (1997) has this entry for "barefaced liar":
Barefaced, "beardless, with no hair upon the face" may have been coined by Shakespeare in A Midsummer Night's Dream, where it is first recorded. Within a half a century or so it came to mean bold, audacious, impudent, or shameless, like many boys, who were barefaced. By 1825 we find "the barefacedness of the lie" recorded, and Harriet Beecher Stowe writes of a barefaced lie in Uncle Tom's Cabin [in 1852].
To similar effect is the entry for "bare-faced lie" in the American Heritage Dictionary of Idioms (1997):
A shameless falsehood....The adjective barefaced means "beardless," and one theory is that in the 1500s this condition was considered brazen in all but the youngest males. By the late 1600s barefaced also meant "brazen" or "bold," the meaning alluded to in this phrase.
But "barefaced lie" goes back much farther than 1825. Here is part of a "Poetical Essay" from The Gentleman's Magazine, volume 5 (October 1735):
Yield, envoy yield; nor longer vainly try/The tim'rous whisper, or the barefac'd lie;/Greatness, superior to thy arts, can view/Its kindred virtues, and admire 'em too.
Other early appearances of the phrase appear in a 1755 translation of Crevier's History of the Roman Emperors and in John Trusler, Life, or the Adventures of William Ramble (1793). Numerous instances appear from 1796 forward.
The earliest association of "bold-faced" and "lie" that I've been able to find in a search of Google Books is this item from Richard Baxter, "The Catechising of Families" (1682):
He that by Equivocation useth unapt and unsuitable Expressions, to deceive him that will misunderstand them, is to be blam'd: But he that will stand openly Bold-faced in a Lye, much more.
Farmer & Henley, Slang and Its Analogues (1890) reports that "bald-faced" meant "having white on the face," and led to two contemporaneous slang terms: "bald-faced shirt," meaning (in cowboy lingo) a white shirt; and "bald-faced stag," meaning a bald-headed man.
One of the earliest metaphorical uses of "bald-faced" is in conjunction not with lying but with impudence. Thus, from Gales & Seaton's Register of Debates in Congress (April 28, 1836), we have this fragment from a speech by a Mr. Moore of New York
And yet, in the teeth of all these facts, in contradiction to all experience, and in defiance of the concurrent testimony of history, our modern aristocracy have the presumption, nay, the bald-faced impudence, to allege that the people have ever a propension to sedition and plunder.
Similar instances of "bald-faced impudence" occur in Punch, volume 17 (1849) and in The Medical Age (January 25, 1895).
Gilbert A. A'Beckett, The Comic History of England (1847) brings "bald-faced" (in the strict sense of "white-faced") and "bare-faced" into close proximity, in this sentence:
The extraordinary hilarity of the bounding hart attracted the attention of Rufus, who drew his bow, but the string broke, and Rufus not having two strings to his bow, called out to Tyrrel to shoot at the bald-faced brute for his bare-faced impudence.
Wentworth & Flexner, Dictionary of American Slang (1960) has entries that cite "barefaced lie" and "bald-faced lie"—but not "bold-faced lie"—as current slang.
My guess is that all three expressions are tangled up historically in various forms of bare-, bald-, and bold-faced behavior, all entailing the same general notion of audacity, impudence, and shamelessness. Under the circumstances, it's hardly surprising that variants of all three words are linked to the words lie and liar.
Another Meaning of "Bald-Face"
Maximillian Schele De Vere, Americanisms: The English of the New World (1872) offers this entry in a chapter on "Cant and Slang":
Bald-face, one of the many slang terms under which bad whiskey passes in the West.
Likewise, John Russell Bartlett, Dictionary of Americanisms, Second Edition (1859):
BALD FACE. Common (penny) whiskey, particularly when it is new; also figuratively and appropriately called Red Eye.
A "bald-face lie" might thus be a lie that someone tells under the influence of bad whiskey; it is far more likely, however, that the drink-related term is unrelated to lying and refers simply to the immediate effect of the liquor on the face of the imbiber.