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Of obscure origin, the obsolete noun clogdogdo is a term of abuse or disparagement, especially for a man’s wife or female relative.

The noun clogdogdo is perhaps composed of the noun clog, of the noun dog, and of an element of uncertain identity (perhaps arbitrary)—the implication being, perhaps, that the person referred to is an encumbrance, as a clog (i.e., a heavy piece of wood) tied to the leg or neck of a dog to impede movement or prevent escape.

This was the explanation suggested by Thomas Lewis Owen Davies in A Supplementary English Glossary (London: George Bell and Sons, 1881) [page 128, column 1]:

Cʟᴏɢᴅᴏɢᴅᴏ, an incumbrance, like a clog tied to a dog. See quotation s.v. Cʟᴏɢ.
A wife is a scurvy clogdogdo, an unlucky thing.—Jonson, Silent Woman, IV. i.

This is the entry that Thomas Lewis Owen Davies was referring to—from A Supplementary English Glossary (London: George Bell and Sons, 1881) [page 128, column 1]:

Cʟᴏɢ, an old-fashioned wooden almanac.
The lineal descendant of that rimstoke was still in use in the middle of England at the close of the seventeenth century, though it was then, says Plot, a sort of antiquity so little known that it had hardly been heard of in the southern parts, and was understood but by few of the gentry in the northern. Clogg was the English name, whether so called from the word log, because they were generally made of wood, and not so commonly of oak or fir as of box; or from the resemblance of the larger ones to the clogs “wherewith we restrain the wild, extravagant, mischievous motions of some of our dogs,” he knew not.—Southey, The Doctor, ch. xc. *

* This refers to The Doctor, &c. (New York: Harper & Brothers, 1836), by the English poet and writer of miscellaneous prose Robert Southey (1774-1843) [Vol. 1, chapter 90, page 145].

The noun clogdogdo was first used—and perhaps coined—by Ben Jonson (cf., below, quotation 1), and later chiefly used in reference to him (cf., below, quotations 2, 3 & 4).

These are, in chronological order, the earliest occurrences of the noun clogdogdo that I have found:

1-: From the following dialogue between Captain Tom Otter and John Daw, in Epicœne, or The Silent Woman, a comedy by the English poet and playwright Benjamin ‘Ben’ Jonson (1572-1637)—as published in The Workes of Benjamin Jonson (London: Printed by William Stansby, 1616) [Act 4, scene 2, page 569]:

Otter: A wife is a sciruy clogdogdo; an vnlucky thing, a very foresaid beare-whelpe, without any good fashion or breeding: mala bestia.
Daw: Why did you marry one then, Captaine?

2-: From Bury-Fair (London: Printed for James Knapton, 1689), a comedy by the English poet and playwright Thomas Shadwell (circa 1640-1692) [Act 3, scene 1, page 39]:

Mr. Oldwit: Where are my Drunkards? where are my Drunkards? You flinchers, you sober Sots! Where is my Iezebel, my Cockatrice, my Clogdogdo, as honest Tom Otter says? A senceless Jade, with her Wit, and her Breeding: She steals away my Drunkards. Old Spouse, Mummy; thou that wrap’st thy self every Night in Sear-cloths!

3-: From The Antiquary (New York: Published by Van Winkle and Wiley, 1816), by the Scottish novelist and poet Walter Scott (1771-1832) [Volume 1, chapter 6, page 58]:

“You are welcome to my symposion, Mr. Lovel; and now let me introduce you to my Clogdogdo’s, as Tom Otter calls them; my unlucky and good-for-nothing womankind—malæ bestiæ, Mr. Lovel. […] You will find them but samples of womankind—But here they be, Mr. Lovel. I present to you, in due order, my most discreet sister Griselda, who disdains the simplicity, as well as patience, annexed to the poor old name of Grizzel; and my most exquisite niece Maria, whose mother was called Mary and sometimes Molly.”

4-: From The Doctor, &c. (London: Longman, Rees, Orme, Brown, Green and Longman, 1837), by the English poet and writer of miscellaneous prose Robert Southey (1774-1843) [Vol. 4, chapter 116, page 133]:

I feel myself bound for the sake of Dr. Dove to vindicate the daughter of his old schoolmaster from a splenetic accusation brought against her by her husband. […]
Gent when he penned that peevish page seems to have thought with Tom Otter, that a wife is a very scurvy clogdogdo!

5-: From a letter to the Editor, by one H. U. Janson, dated Exeter, September 1846, published in Woolmer’s Exeter and Plymouth Gazette for the West of England; Or, Public Advertiser for Devon, Cornwall, Somerset, and Dorset (Exeter, Devon, England) of Saturday 12^th September 1846 [page 4, column 2]—I have not discovered whom “Mr. Oldbuck” refers to:

I have much pleasure in drawing the public attention to […] the science of Pʜᴏɴᴇᴛɪᴄꜱ; towards which, people will act in the self-same manner,—that is to say, they will accept or reject it, in exact proportion to the good or bad construction of their own minds; for mankind may, in this respect, be divided into two great classes,—those who are advancing with their age; and those who from carelessness, indolence, and stupidity, are ever hanging back, and telling us that the old system of things “does well enough!” It is to the former class that I address myself—the latter are merely the drag-chains of society—the “clogdogdoes,” as Mr. Oldbuck says; and quite unworthy of notice.
[…]
I have long been secretly occupied with this very interesting and important subject; watching its progress under the able direction of the ingenious Mr. Isaac Pitman, of Bath, who, aided and abetted by other individuals of high talents and acquirements, has, after three years of the most laborious exertion, succeeded in bringing the new Phonetic Alphabet to a very high state of perfection. As this is merely an introductory, or “leading article,” I will not occupy more of your space at present, than just to observe, that I have procured, from the Bath Phonotypic Fount, the new alphabet, together with a short specimen of Phonetic Printing—or printing by sound. It will immediately take with the individuals of the first class, to whom I have alluded—and to the “clogdogdoes” I have merely to observe that although to them it will doubtless look “very queer,” yet it does not look more so than the “oulde” English of the time of Chaucer, &c.,—which, doubtless, the “clogdogdoes” of that age thought was “quite well enough.”