As in, a word signifying 'a little' - used in common vernacular in England, and possibly elsewhere.
As Barrie and Peter have already stated, the OED and Etymonline leave it as being perhaps from Scots smitch, without further clarifying what that is or where it comes from. With no authoritarian clues to guide us, I will indulge in some unfounded (but, I think, reasonable) speculation:
My personal guess would be that this is one of a group of words in Irish and Scottish Gaelic (since my knowledge of Scottish is quite rudimentary, I’ll refer only to the Irish forms here—it’s quite a safe bet that most, if not all, of these also exist in some form or other in Scottish) that refer to something seen as tiny and insignificant, either something verbal or a physical object.
These words all start with sm- plus a vowel usually either /ɪ/ or /ɐ~ə/ and one more consonant. Most of the words are monosyllabic, with this consonant being then final, but some have an extra syllable, usually /ə/, /əɾ/, or /əɫ/ (these are all common nominal endings and are not part of the root of the word—the latter two are non-productive diminutives). There appear to be three groups (all words taken from Foclóir Uí Dhuinnín):
Final g = verbal
Smiog /smɪɡ/, smiogal /ˈsmɪɡəɫ/ ‘tittle, word, peep’; ní raibh smiog(al) as ‘he didn’t say a peep’
Final d = verbal or physical
Smid /smɪdʒ/, smaid /smɐdʒ/ ‘tittle, word, peep’; ní raibh smid ná smaid aige ‘he had nothing to say’, ‘he was taciturnly silent’
Smioda /ˈsmɪdə/ ‘piece, chunk’
Smiodar /ˈsmɪdəɾ/ ‘scrap, piece, fragment’; ní raibh smiodar den chíste romham ‘not a scrap of the cake was [left] in front of me’
Smeadar /ˈsmɐdəɾ/ ‘mess, jumble, dirty’; tá mo lámha ’na smeadar agam ‘my hands are all messy/dirty’
Final t = physical
Smiot /smɪt/ ‘stump, lump’
Smut /smɐt/, smuit /smˠɪtʃ/ ‘stump, anything short or stumpy, piece of something, nose [esp. of animal]’; tá smu(i)t den cheart agat ‘you’re partly right’ (lit. ‘you’ve got a stump of right’); tabhair domh smu(i)t de sin ‘give me a drop/bit/crumb of that’
Smuta /ˈsmɐtə/ ‘bit, (small) portion’; smuta gáire ‘slight smile’ (lit. ‘a small bit of a smile’)
The most common way to form a diminutive in Irish (and Scottish) is to add one of the suffixes -ín, -(e)án, or -óg/-eog at the end; the first of these is by far the most common and productive one. It causes a preceding consonant—i.e., the final consonant in the root or the final r/l in the suffix, if there is one—to become slender if it isn’t already (see Irish phonology for what broad and slender consonants are). Sometimes, this slenderisation is continued backwards to the root-final consonant even if there is a suffix, but not always; similarly, the schwa in the -al/-ar suffixes is sometimes deleted, sometimes not.
All the words listed above can be turned into diminutives very easily with this suffix, and quite a few of them would either coincide in Irish or be so similar that they would certainly coincide if borrowed into English:
Smiog > smigín; smiogal > smiogailín/smigilín/smiglín
Smid and smioda > smidín; smaid > smaidín
Smiodar and smeadar > smidirín (or sometimes smidrín)
Smiot > smitín
Smut, smuta, and smuit > smuitín
Of these, a few are common enough to have entered the dictionaries: smidirín ‘tiny piece/scrap/fragment’ (this is the source of the English word ‘smithereens’, as in ‘blow something to smithereens’); and smitín ‘rap, smart blow as with a stump of wood’.
Of the others, I myself have certainly heard people using smidín and smuitín in regular speech in Irish to refer to little scraps of paper, pieces of dust, flecks of paints, specks of dandruff, smudges—things like that.
My guess would be that ‘smidgen’ (and perhaps also ‘smudge’) in English comes from one or more of these diminutives and their source words. Both ‘smidgen’ and ‘smudge’ appear to have existed originally both with a final /dʒ/ and a final /tʃ/, which fits perfectly with being from a mixture of these words that also vacillate between final d and final t.
It appears from a bit of Googling that this Reader’s Digest PDF gives a similar etymology, so I’m not the first to advance this theory after all, it seems.