Attested in the 15^th century, the noun trivet designates a three-footed stand, particularly one for a pot, kettle or other vessel placed over a fire for cooking or heating something.

(The noun trivet seems to be derived from the Latin tripēs/tripĕd-, meaning three-footed, from tri-, three, and pēs/pĕd-, foot.)

The phrase (as) right as a trivet means thoroughly or perfectly right, in reference to a trivet’s always standing firm on its three feet.

These are the earliest occurrences of the phrase (as) right as a trivet that I have found, in chronological order—the first two refer to boxing:

1-: From The Observer (London, England) of Sunday 13^th June 1824—on Tuesday 8^th June 1824, Tom Spring had defeated Jack Langan in a bare-knuckle boxing match:

On Wednesday, a rumour was circulated through London that Langan had died, in consequence of the severity of his punishment, and that Spring, and the seconds and bottle-holders, were forthwith to be lumbered for his murder. […] As on every subject connected with Sporting events, the matter became a subject of betting, & odds were laid that Langan was “as right as a trivot [sic],” and would fight within six months.

2-: From the Weekly Dispatch (London, England) of Sunday 13^th July 1828—a bare-knuckle boxing match between Bishop Sharpe and Alec Reed was to take place on Tuesday 15^th July 1828:

The Fancy Scrap-Book. No. 15.
DIALOGUE BETWEEN ALEC REED AND BISHOP SHARPE, ON THEIR APPROACHING MILL.

In training for action, and keen for the fray,
While surveying the beauties of nature,
Sharpe, the Woolwich Will Watch, in his rounds t’other day,
Met Alec, the Chelsea translator. (1)
Like heroes who combat for glory—and gain (2)
No hostility sullied their greeting,
And what passed between them, I could not refrain
From then noting, and now from repeating.

R.—To have a fresh shy for my luck in the ring,
In vain, Sharpe, too long I had sought you;
Through my friends, I, at last, have accomplished the thing.
And I think, for this once, I have caught you:
Next Tuesday will tell your supporters a tale,
For, if I know my powers as a fighter,
The Chelsea preserver of soles will not fail
To eclipse the episcopal mitre.

In slap up condition to combat I’ll come,
Prepared either to take it or give it,
Not an ounce in my bones you’ll discover of gum,
My frame is as right as a trivet.
[…]

(1) Translator, alias a shoemaker. Alec’s first fighting cognomen was the Chelsea Snob ¹. It has been said that nothing but a Bishop gains by translating. Query—Is not the sole (of a shoe) also benefited by it? As to what the other soul may gain or lose by translation, this deponent sayeth not.
(2) And what heroes do not? War, like virtue, is, I apprehend, but rarely loved for its own sake only. It is this that makes the cant about pugilists fighting for money so sickening. Does not every soldier, however brave or patriotic, do the same thing; from the private with his daily stipend of fourteen pence, to the Noble Duke, who by the trifling receipt of somewhat more than half a million, in one shape or other, was saved from reaping laurels altogether barren? As the old song has it—
“The soldier to fight, without pay,
To fight would be wonderful lazy.”

¹ The original meaning of the noun snob was a shoemaker or cobbler.

3-: From The Morning Chronicle (London, England) of Thursday 26^th March 1829:

March of Intellect.—The following is a verbatim et literatim copy, in St. Giles’s Greek ², of a card issued by a breeches-maker, in the Edgware-road:
“C. Hammond, carver of slap kicksies, trotter cases, mud pipes, and boot kivers, Connaught-terrace, Edgware-road.—A report being in circulation, that having renewed the lease of my little Crib, and at a great expense enlarged my shop, by attaching the adjoining premises, that I intend to lay it on my customers pretty thick, or, in other words, to tip it them pretty stiff in my charges—on the contrary, I am enabled to lay in a larger stock, having more room to stow it, and having purchased largely with the ready stuff, I can build kicksies, &c., for very modest figures. The following young prices will speak aloud:—
“A pair of out-and-out Kersey Kicksies, got up slap, with pearl buttons and leather strings, or artful buttons, &c.—One pound and three-quarters.
[…]
“A pair of cloth Mud Pipes, with unbleached pearl buttons, commonly called horn.—A Canary and a Bull.
“N.B.—Not over nice to taking a bit of Stiff, if the Barring is as right as a Trivet.”

² The phrase St. Giles’s Greek designates criminals’ slang. Named after the church of St. Giles-in-the-Fields, St. Giles is a parish and district in London; it was noted for its high proportion of criminal inhabitants—cf. origin of the phrase ‘it’s Greek to me’.

4-: From the Morning Advertiser (London, England) of Tuesday 28^th April 1829:

CURRENCY.
LETTER IV.
To the Editor of the Morning Advertiser.

Sir,—It is a very common practice when the patient of a chronic disease is almost at the threshold of the tomb, and shows indubitable signs that his dissolution can’t be far off, for either the sufferer or his friends to doubt the judgment of the regular family physician, and (although, mayhap, his lenient remedies had hitherto kept the patient upon his legs, and the chances may be that he must know his patient’s case and “constitution” better than a stranger) to call in another batch of doctors, nay, quacks and all, “for hope still lingers till the dying gasp,” but there the man must die; and it is not an uncommon case for the new batch of doctors and their nostrums to hasten his fate.
So it is, Sir, with a deranged financial system, when once it shows symptoms of derangement and declension, every financial quack will bring out his nostrum. Although we have already disposed of Sir John Sinclair, Bart., and Privy Councillor, of Mr. Burgess, &c., &c., let us not disrespectfully pass over in silence the remedy of another great currency philosopher, Sir Thos. Lethbridge, Bart., for he too was “among the prophets,” and had his remedy! “We ought,” said the Baronet, “to adopt the Scotch system of banking, which has prospered so long; (so it has,) and then we should be all right;” aye, right as a trivet when a burnt-out log falls into cinders from under one of its feet.