Where did "snuck" come from?
Solution 1:
It's just an incorrect tense construction of the verb that passed into common usage.
Verb tensing requiring a change in vowel is among the hardest area of grammar to create hard and fast rules for. The majority of past participles just add "ed" (or sometimes just "d"), such as walked, soaked, etc.
However, when that doesn't work there are really no good rules to say how the vowel should change. There are also some distinctions originally made between various past tenses that have been lost in colloquy; for instance, different conjugations ("it stank", but "I stunk") or between various past tense constructions resulting in passive/active voice differences ("I sang", but "the song was sung").
"Sneak" is a verb that is technically regular in past participle formation ( add -ed to form "sneaked"), but because it is phonetically similar to some exceptions that change the present tense vowel to "u" (stunk, sung, sunk), the vowel change is becoming acceptable for this word as well. Gotta love the organic nature of human language.
EDIT: Further exercise of my "Google-fu" has brought up this blog, which in turn references this BBC article. It appears that the word "sneak" is the latest of a series of verbs that have undergone "weak to strong drift".
To explain (hopefully consicely): early Anglo-Saxon language categorized verbs in several classes depending on how their various forms were constructed. This ancient system is where we get such peculiarities of tensing as bring/brought, think/thought, see/saw, fly/flew, and sing/sang/sung. Over time, the various verb classes coalesced into two: verbs that formed the past participle by simply adding "-d"/"-ed" were "weak", and verbs that formed the past participle any other way were "strong".
It is unusual but not unheard of for a verb's conjugation and tensing rules to change. However, when it does, it's usually "strong to weak"; conjugation and participle construction are simplified from the complex vowel-changing rules to easy suffixing. The word "glide" used to have the past participle "glad", which has been completely abolished in modern usage in favor of the "weak" construction "glided".
However, "sneak", and words that have gone before it like "dig", "string" and "dive", went the other way; they went from simply adding the suffix to changing the vowel. "Digged" became "dug", "stringed" became "strung", and "dived" became "dove". In this same way, "sneaked" is becoming "snuck"; the word "snuck" is already generally accepted as "sneak"'s past participle in most of the English-speaking world except for Britain. Eventually, it is thought, those tea-timers will give in to the pressures of the colonies.
Solution 2:
Snuck, according to an 1890 Dialect Notes, originates from Western Ohio. This is also the third oldest reference I found.
Ballou's monthly magazine, Volumes 53-54, 1881:
Well, sir, your boy Aleck got a straw, snuck up behind a sorrel mule, tickled him on the hind leg, and" — The lady gave a wild shriek, and started for the door. — "and the blamed critter never lifted a hoof. Never so much as switched
The Southern Bivouac, Volume 6, 1887:
One bright, golden, delicious afternoon in the latter part of May, Jim left the patch where he had been hard at work all day, and "snuck een" to his cabin by the back way. He proceeded hastily to doff his everyday clothes and don his ...
Dialect notes, Volume 1, 1890:
snuck, v., trans, and intrans. : to sneak. " He snucked that," "he snucked up to it." Western Ohio (Cognate of snug ? J.M.H.)
Snuck subsequently started sneaking in, slowly, and has snuck its way surely but steadily ever since:
Source
Solution 3:
I found a comprehensive discussion of the word here.
Usage has changed...
In present-day English, snuck is extremely widespread throughout the country, even among educated speakers, and in the speech of younger people it is the dominant form. Many younger speakers are unaware that sneaked exists, or think that it sounds as wrong as many older speakers think snuck does. It is absolutely incorrect to say that snuck is "nonstandard," as one current usage book does, or even that it's "wrong," as many people believe. It is not even "informal" or "jocose," as one reasonably up-to-date book says. Snuck is fully standard in American English...
Many people object to snuck, but that has been changing and is likely to change further as younger snuck-ful speakers age and enter the mainstream. Sneaked is still somewhat more common in print, but that is probably reflective of the fact that sneaked is favored by more conservative editors and copy editors.
Here is some history, from Merriam-Webster, along with the following observation:
...sometime in the late 19th century a variant irregular form, snuck, began to appear in the United States
"...an' den snuck home" -The Lantern (New Orleans), 17 Dec. 1887
"Dock Knowital he Snuck Out the room an' Disappeared" -Frank W. Sage, D.D.S., Dental Digest, November 1902
"...I snuck off down the street and got something to eat" -Ring Lardner, You Know Me Al, 1916
...
But where does snuck come from? We don't know; it is as mysterious as the origin of sneak itself. The only evidence we have suggests snuck is a late 19th-century North American innovation. One theory suggests it may have been a survival in some obscure northern English or Scottish dialect brought here by settlers. It is tempting to trace it form Old English snican; Old English strican of the same class of strong verbs gave us strike and struck; there is at least one surviving instance of Middle English snike, around 1240, which is clearly derived from snican. But no evidence survives to connect either sneak or snuck conclusively to snican and snike. It is a long time from 1240 to 1887.
I would say that, given the indefiniteness of this case, it seems plausible that the form strike/struck leaked into sneak/sneaked, giving sneak/snuck.
Much more speculation would appear to be useless.