Etymological discussions of 'geezer' in various dictionaries
The primary link offered in the accepted (and highest-voted) answer is now longer functional and the other links cited (from Dictionary.com) are rather meager, so I thought it might be useful to list what several sources have to say about the origin of the word geezer. There is strong agreement that (as both Gigili and Thursagen report in their answers) geezer derives from guise.
From John Ayto, Dictionary of Word Origins (1990):
geezer {19[th century]} Originally, a geezer seems to have been 'someone who went around in disguise.' The word probably represents a dialectal pronunciation of the now obsolete guiser 'someone wearing a masquerade as part of a performance, mummer.' This was a derivative of guise {13[th century]}, which, together with disguise {14[th century]}, goes back ultimately to prehistoric Germanic * wīsōn, ancestor of archaic English wise 'manner.'
This is the excerpt cited in Gigili's answer.
From Ernest Weekley, An Etymological Dictionary of Modern English, volume 1 (1921):
geezer {slang}. Dial[ectal] form of obs[olete] guiser, mummer. See guise. [Cited example:] Nice old geezer with a nasty cough [—](Albert Chevalier).
...
guise. F[rench], O[ld] H[igh] G[erman]. wīsa (weise), manner, whence also It[alian] guisa. ... Sense of style, costume, survives in disguise, in the guise of, etc., and dial[ectal] guiser, mummer, whence geezer.
From Eric Partridge, A Short Etymological Dictionary of Modern English (1958):
geezer (sl[ang]), old person: prob from obs guiser, an actor (see GUISE), but perh introduced by sailors and soldiers from Malta (? gisem, the human body, hence a man).
From Robert Hendrickson, Encyclopedia of Word and Phrase Origins, fourth edition (2008):
old geezer. This expression is something of a redundancy because geezer itself means an eccentric old man. Geezer is a British dialect word for a mummer or masquerader who wore a disguise and often acted erratically.
Early instances of 'geezer' and its variants in print
The answer posted by ghoppe contains part of the entry for geezer from J.S. Farmer & W.E. Henley, Slang and Its Analogues Past and Present, volume 3 (1893):
Geezer, subs, (popular). — An appellation, sometimes, but not necessarily, of derision and contempt; applied to both sexes, but generally to women. Usually, OLD GEEZER. For synonyms, see WITCH. [Cited examples:] 1885. Truth about the Stage, p. 16. If we wake up the old GEEZERS we shall get notice to quit without compensation[.] 1886. Broadside Ballad, 'Her Mother's Got the Hump.' This frizzle-headed old GEEZER had a chin on her as rough—well as rough as her family, and they're rough 'uns. 1890. A Chevalier, 'Knocked 'Em Dead on the Old Kent Road.' Nice old GEEZER with a nasty cough. 1892. Anstey, Voces Populi, p. 82. Our old GEESER's producin' the customary amount o' sensation.
Joseph Wright, The English Dialect Dictionary, volume 2 (1900) similarly notes that a geezer may be male or female:
GEEZER, sb. ... Slang. A queer character, a strangely acting person; an old man. See Guiser. [Cited examples:] N[orthumberland] She's a geezer that neyber abuv, WILSON Tyneside Sngs. (1890) 202; N[orthumberland] He's a reg'lar geezer, that'n is (R.O. H[ESLOP, Glossary of Words Used in the County of Northumberland (1892–1894)]) N[orthumberland;] w[est] Y[orkshire] Applied to a female whosw habits are of a mean and despicable character, Leeds Merc. Suppl. (July 15, 1893). Lan[cashire] (F.R. C[HALMERS]) I[sle of] Ma[n] Tell the old geezer I'll be going to chapel reg'lar, CAINE, Manxman (1894) pt. 1.vi. Lon[don] Master, 'governor' (F.R. C[HALMERS]), Slang, FARMER.
John Wilson, Tales of the Borders, and of Scotland (1835/1857) uses the spelling guyser to refer to the (male) maskers who roved about on Hogmanay night demanding the treat of pease-bannock and threatening otherwise the trick of "squalling bagpipe" and smoke from the "smeikin horn":
The evening set in and the witching hour—the keystone of night's black arch, twelve o'clock—was approaching. To go to bed on such an occasion, would have been held no better than for the jolly toper to shirk his bicker, a lover to eschew the trysting thorn, or a warrior to fly the scene of his country's glory; neither would it have been safe, for no good guyser of the old school would take the excuse of being in bed in lieu of the buttered pease-bannock—the true hogmanay cake, to which he was entitled by "the auld use and wont" of Scotland; ...
A Google Books search does find at least one example where a "geeser" is a woman. From Judy, Or the London Serio-Comic Journal (August 13, 1884):
"Heap tender epithets upon me," warbled the bride; "hurl as many as you like at me, I shall not flinch much, for, though an arid imperious spirit, I love you." "You are a geeser," said her husband, who had married for money. "A geeser," exclaimed the elderly better half, with eagerness, "what's that?" "Oh, confound it! how your education must have been neglected in early youth," said the lord of creation in a lofty manner; and then, after an explanation, they had quit a little tiff.
An Elephind search finds an instance of "old Geezer" from 1873. From "Chips by a Sandalwood Cutter," in the [Fremantle, West Australia] Herald (September 13, 1873), narrated by an unidentified male character/newspaperman:
"Well" says I, "there ain't no denyin' there's a good deal of truth in what Lord Kimberley says, but I ain't a goin' to throw a stone at a thrashed dog—and so I won't publish that speech you made the other evenin' at the ministerial dinner."
"Cuss you" says he, "you mischeevious old Geezer, was you a goin' to dish up in you dirty bush-pie fashion the remarks I made at that dinner?"
In the United States, geezer (without the modifier old) shows up in a ditty reproduced in the Memphis [Tennessee] Daily Appeal (February 10, 1881):
The mercury touched zero;/ The day was just a sneezer,/ And anxiously, the toper/ Was hunting for a geezer./ He found a flask of benzine;/ He thought that it was gin;/ He quickly did invert it,/ And scooped the contents in.
This, however, is geezer in the unrelated sense of "a drink of whiskey or other alcohol, which J.E. Lighter, Random House Historical Dictionary of American Slang (1994) traces to Lincolnshire, England in the year 1866. It appears that this geezer may have to have reached the United States earlier than the eccentric old man did.
The earliest instance of "old geezer" in a Google Books search, however, refers to a woman. From "Splashes from our Gridiron," in The Angler's Journal (January 16, 1886):
Mr. J. Dallas has a sympathetic part in Queen Elizabeth, and, moreover, has the moral courage to give a true, if somewhat unflattering, portrayal of that eccentric female ; for in spite of what we learnt as children in our for the most part one-sidedly-written history-books, we all know what a foolish, spiteful, vain old geezer she was.
But a male old geezer appears less than two years later in "Now, Which Is It, Please," in Judy, Or the London Serio-comic Journal (November 23, 1887):
"That's him!" cried Chawley ; " at the back—/ A regular old Blazer!/ I knows him, s'help me, it's a fac'—/ It's Alderman De Kay-ser!"/ "D'ye mean," said 'Arry, nothing slow,/ " That pompous-like 'old geezer'?/ I've been before him—I should know—/ That's Alderman De Kee-ser!"/ ...
Another early instance of "old geezer" appears in a bit of doggerel from J.H. Coyne, identified as "a member of the Chicago Press Club," in Tobacco: An Illustrated Weekly Journal for the Wholesale and Retail (October 18, 1889):
There was an old Geezer,/ And he had a wooden leg,/ And he never had Te Baky,/ Eksep wot 'e kud beg./ .../ Says the old Geezer,/ Will you give me a chew,/ Says Uncle Billy Daniels/ I'll be hanged if I do; ...
It remains a mystery to me how the original term guiser, which seems to have had a strongly male implication, came to be applied (as geezer or geeser) quite frequently to eccentric old women (as well as to old men), and why subsequently it came to be associated exclusively with old men.