Barbara Kipfer & Robert Chapman, Dictionary of American Slang, fourth edition (2007), offers three relevant definitions of flake as a noun:
flake ... n 1 An eccentric person, esp a colorful individualist; BIRD : what is known in the trade as a flake, a kook, or a clubhouse lawyer [quotation attributed to Christopher Lehmann-Haupt] / Users and flakes clung to her [quotation attributed to New York Magazine] (1950s+ Baseball) 2 The quality of flamboyant individualism : The Yankees have acquired an amount of "flake" [quotation attributed to Leonard Koppett, New York Times, April 26, 1964] (1960s+ Baseball) 3 A stupid, erratic person; RETARD (1960s+ Teenagers)
and three relevant definitions of flaky/flakey:
flaky or flakey adj 1 Colorfully eccentric; buoyantly individualistic (1960s+ Baseball) 2 Insane; SCREWY, WACKY : a flaky old professor, a snake expert [quotation attributed to New York Magazine] (1960s+) 3 Disoriented; barely conscious; dizzy : He played the last 23 minutes of the game in a condition that was described as "flaky" and "fuzzy" [quotation attributed to New York Times] (1960s+)
The Leonard Koppett quotation used to illustrate definition 2 of flake above strays somewhat from the wording it had in Harold Wentworth & Stuart Flexner, Dictionary of American Slang, second supplemented edition (1975). Here is the wording given in the earlier edition of the dictionary:
"In the last three years, the Yankees [New York baseball team] have acquired ... [an] amount of "flake" in the person of such young players as Joe Pepitone, Jim Bouton, Phil Linz...."
The earliest use of the noun flake to refer to an offbeat and unpredictable person that I have seen reported is cited in this entry from The Dickson Baseball Encyclopedia, third edition (2011):
flake An odd or eccentric player; a kidder or comic; a kook. Famous baseball flakes include Jackie Brandt (the "original" flake), Phil Linz, Denny McClain, Bill "Spaceman" Lee, Mark Fidrych, Al "The Md Hungaria" Hrabosky, Ross Grimsley, and Jay Johnstone. ... USAGE NOTE. The term carries a certain element of endearment and tends to be applied to likable, but not always reliable, characters. ETYMOLOGY. By the late 1950s, the term "flakey" began to replace the beatnik term "kook" for an oddball character. According to Joseph McBride (High & Inside, 1980), a San Francisco Giants teammate in 1957 gave offbeat outfielder Jackie Brandt the nickname “Flakey” because “things seem to flake off his mind and disappear.” The adjective became a noun ("flake") by the early 1960s. Maury Allen (Bo: Pitching and Wooing, 1973) claimed that Brandt's St. Louis Cardinals teammate Wally Moon created the term in 1956: "Moon suggested that Brandt was so wild his brains were falling out of his head, flaking off his body, hence, a flake."
So according to Dickson, flake originally arose in connection with the outfielder Jackie Brandt in 1956 or 1957, although Brandt's nickname was "Flakey" not "Flake." Dickson also notes that, outside baseball, the word flaky eventually acquired a less affectionate sense, "as when President Ronald Reagan said that Col. Mu'ammar al-Gadhafi was 'flaky,' thereby implying that the Libyan strongman was strange and possibly not sane."
Jim Brosnan, The Long Season (1960), a diary of his 1959 season with the St. Louis Cardinals and Cincinnati Reds, offers what may be the first recoded definition of flakey, as a glossary entry:
Flakey A character, or characteristic, of eccentric or unusual behavior, verbal or physical, on or off the ball field. Any ballplayer who is considered hard to figure out is called "flakey."
But in Brosnan's book, the term comes up as a nickname not for Jackie Brandt (who was then with the San Francisco Giants) but Joe Cunningham (a Cardinals first baseman/outfielder), from an entry dated August 23, 1959:
"He [Cunningham] and [Eddie] Kasko make a real funny pair," said [Reds pitching coach Cot] Deal. "They could beat [Dean] Martin and [Jerry] Lewis when they get started."
"How come Cunningham is called Flakey, Cot? When did that start?"
"People that don't know him call him that. He's got an oddball sense of humor, and at the same time he's so damn serious about baseball that you'd think he's two different people. 'Flakey' means people don't understand you. ..."
I agree with Hot Licks's conclusion (in a comment above) that the core characteristic of a flake isn't unreliability but unpredictability, which is why it can be used to refer to someone who does endearingly odd things or to someone who does irrationally harmful things.