The phrase conspicuous by one’s (or its, etc.) absence means obviously not present where one (or it, etc.) should be.

The British statesman John Russell (1792-1878) is generally—but erroneously—credited with coining this phrase in his Address to the Electors of the City of London, published on Wednesday 6^th April 1859.

This is the relevant passage from this address, as published in The Spectator. A Weekly Journal of News, Politics, Literature, and Science (London: Published by Joseph Clayton) of Saturday 9^th April 1859:

Lord John Russell’s address is as follows—
“Gentlemen—A dissolution of Parliament has been announced, and I have again to solicit a renewal of your confidence.
“Her Majesty’s Ministers, early in the session, introduced a so-called Reform Bill. Among the defects of the bill, which were numerous, one provision was conspicuous by its presence, and one by its absence; a non-resident right of voting in boroughs was introduced. By this means fictitious votes would have been created, and the abuses so prevalent before the enactment of the Reform Bill would have been revived. A small borough is not necessarily corrupt or servile, but under this bill it would easily have become one or the other. By the same clause freeholders whose property was in boroughs were deprived of their votes for the county: life interests only were respected. In this manner the counties were to be deprived of their most liberal element, and the addition of ten-pound occupiers was treated as a poison to which an antidote was required.
“The absence of any provision to reduce the franchise in boroughs and the hard line of separation thus left between the middle orders and those who earn their livelihood by manual labour would have tended to foster discontent, and make a war of classes.”

However, although John Russell popularised the phrase, it had already been used on two occasions at least:

1-: By the Earl of Ellesmere *, according to the account of a debate at the House of Lords, published The Standard (London, England) of Tuesday 18^th May 1847:

FACTORIES BILL.

The Earl of Ellesmere having presented several petitions in favour of this bill, said that he now rose to discharge the duty he had undertaken of moving the second reading of the bill for limiting the hours of labour in factories. He was induced to come forward on this occasion, in consequence of the support he had received from a very large and respectable body of persons, who were deeply interested in the fate of this measure. He was also standing there at the earnest desire of a noble friend of his, of whom he might say, in reference to this subject, what was said of a celebrated Roman on the occasion of his absence from a great procession that had taken place to his honour— that he was more conspicuous by his absence from either house of parliament than by his presence on such an occasion.

* The British politician Francis Egerton (1800-1857) was the 1^st Earl of Ellesmere.

2-: As conspicuous from its absence in the following from The Globe and Traveller (London, England) of Thursday 25^th May 1854:

A Peace Member has just been teaching the Peace Society, at its annual meeting, the true peace policy; and as it is never too late to learn, we trust that the respectable association will henceforward come over to the true peace interest. Thirty-eight years ago the Society got itself established, and promulgated its one idea, that the whole duty of the politician was the preservation of peace; in other words, that England, under no circumstances, should war any more. […]
Yesterday the Peace Society held its anniversary meeting, with Mr. Charles Hindley as chairman […]; but they only met to sing a requiem over their failure; and to express their amazement how, after a course of lectures for thirty-eight years from peace orators, the wicked world could ever again go to war. Mr. Hindley piteously confessed that he had indulged in the fond hope that no Christian would dissent from the monstrous doctrine, that “it is wrong to take up the sword in vindication of what are called national rights.” He had thought that it would be impossible again to agitate the people in favour of war. But, he proceeded to confess that he had received a letter from one of his constituents, “a man of great ability, and a member of the Christian Church”—who writes in this good English style:—
“I see you are to preside at a Peace meeting. I hope you will give Russia what she deserves. I would blow up all her ships and soldiers. I am a man of peace, but this is the time for fighting.”
As the statue of Brutus was the more conspicuous from its absence, so the Christian gentleman of Ashton-under-Lyne was the most conspicuous, and the best peace-man at that meeting. […] We trust that the Christian gentleman of Ashton-under-Lyne is only the first recruit of an important corps. Now that the correct policy has been ascertained, we may very well anticipate the pleasure of being joined by the Peace Society Volunteer Regiment.

John Russell in 1859 [cf. footnote], the Earl of Ellesmere in 1847 and the anonymous author of the article published in The Globe and Traveller of 25^th May 1854 were alluding to the following passage from the Annals, Book III, 76, by the Roman historian Tacitus (circa 56-circa 120 AD):

Junia too, the niece of Cato, wife of Caius Cassius and sister of Marcus Brutus, died this year, the sixty-fourth after the battle of Philippi. Her will was the theme of much popular criticism, for, with her vast wealth, after having honourably mentioned almost every nobleman by name, she passed over the emperor. Tiberius took the omission graciously, and did not forbid a panegyric before the Rostra with the other customary funeral honours. The busts of twenty most illustrious families were borne in the procession, with the names of Manlius, Quinctius, and others of equal rank. But Cassius and Brutus outshone them all, from the very fact that their likenesses were not to be seen.—From Annals of Tacitus. Translated into English, with Notes and Maps. By Alfred John Church and William Jackson Brodribb (London: Macmillan and Co., 1876).
     original text:
Et Iunia sexagesimo quarto post Philippensem aciem anno supremum diem explevit, Catone avunculo genita, C. Cassii uxor, M. Bruti soror. testamentum eius multo apud vulgum rumore fuit, quia in magnis opibus cum ferme cunctos proceres cum honore nominavisset Caesarem omisit. quod civiliter acceptum neque prohibuit quo minus laudatione pro rostris ceterisque sollemnibus funus cohonestaretur. viginti clarissimarum familiarum imagines antelatae sunt, Manlii, Quinctii aliaque eiusdem nobilitatis nomina. sed praefulgebant Cassius atque Brutus eo ipso quod effigies eorum non visebantur.—Source: The Latin Library.

Note: On Saturday 16^th April 1859, The Standard (London, England) reported that during a meeting of the Liberal electors of the City of London, held at the London Tavern on Friday 15^th April 1859, John Russell had explained the origin of the expression conspicuous by its absence:

Lord John Russell, who was very warmly received, said—[…] Such was one part of the bill. There was another. The term which I applied to it has been called a misnomer, and by some a bull—I called it a provision “conspicuous by its absence;” but I may be permitted to acknowledge that the phrase was not original, but was taken from one of the greatest historians of antiquity.