I covered this story in possibly extreme detail here, including discussion of the sources.

The tl;dr is that the sultan did donate £1,000; the story that he wanted to donate far more can be traced back to Charles Mackay's supplement to W. Cooke Taylor's three volume Life and Times of Sir Robert Peel. This is a contemporary source, published in 1851, and it was written by a noted journalist who worked for, and went on to edit, the highly influential Illustrated London News. Mackay's brief passage contains in essence the full version of all modern accounts of Abdülmecid's contribution: the offer of £10,000, British objection to the offer, and its revision to a more fitting total. Mackay also seems to have been the earliest author to mention the idea of Turkish aid ships, though it will be noted that his account implies that the idea of sending them was dropped when the Sultan's gift was scaled back to £1,000.

I have done some brief research into Mackay and his sources without really being satisfied that I can account for where he got this extraordinary information. He was a well-connected journalist, but there is nothing in the brief life contributed to the Oxford Dictionary of National Biography by Angus Calder to suggest that those connections extended to the world of politics. Mackay was, moreover, a lifelong liberal, while Peel had led the Tories, which must imply that he was unlikely to have been especially close to many figures in the government of the time.

It seems reasonable to suppose that Mackay came upon his information while researching his contribution to the Life and Times, but how he got that job, and who he spoke to in completing it, is not clear from the volume itself, which is neither footnoted nor referenced. It does seem fairly likely that Mackay was personally acquainted with his predecessor, Taylor, who died around this time; both men were prominent campaigners against the protectionist Corn Laws that Peel abolished. If so, he may have inherited papers and other information, and it is interesting to note, in this respect, that Taylor was an Irishman who, in 1847, was offered a position by the British Viceroy in Dublin, Lord Clarendon. He was subsequently employed as Clarendon's personal statistician. It is not impossible that this job placed Taylor in a position to know a good deal about the internal workings of the relief effort at precisely the time that Abdülmecid became involved in it.

A second source, as you indicate, is a book on the Irish nationalist WJ O'Neill Daunt, A Life Spent for Ireland, a selection from Daunt's journals that was edited by the old nationalist's daughter and published in London shortly after his death in 1894. Here we're told that the Sultan appears to have learned of the famine from his Irish physician, Justin McCarthy of Cork. Daunt had his account from McCarthy's son, who was in Scotland in 1853 and met him there. According to Daunt's journal, their conversation took place on 18 January of that year:

McCarthy... told me that the Sultan had intended to give £10,000 to the famine-stricken Irish, but was deterred by the English ambassador, Lord Cowley, as Her Majesty, who had only subscribed £1000, would have been annoyed had a foreign sovereign given a larger sum...

So unless you take the view that the entry was a hoax, created later, the story dates to about six years after the fact, rather than 40. Moreover, Daunt cannot have been Mackay's source, for he met Justin McCarthy's son two years after the final volume of Peel's Life and Times was published. It is possible that Mackay was McCarthy's, but the incidental detail and the addition of Lord Cowley (who was the Duke of Wellington's nephew and certainly was ambassador to the Sublime Porte at this point) in Daunt's account perhaps suggests that this was not the case. It is possible to suggest, therefore, that the two versions of the tale are probably independent and hence corroborate each other. However, any corroboration only extends to the portion of the story in which the sultan originally wanted to make a larger donation, and was dissuaded – not to the matter of the ships.

Here, finally, there is the matter of the shipping records of Drogheda, the small port to the north of Dublin at which, Irish tradition insists, Abdülmecid's cargoes were unloaded. These records are not available for study, but they are summarised by both the local newspapers of the day, the Argus and the Conservative; shipping news was a staple of most periodicals at this time, since it was vital to the fortunes of the merchants and investors who made up a high proportion of their readers. They suggest that ships hailing from anywhere but Ireland and Britain were rarities in famine-era Drogheda – a point that would certainly explain why the Sultan's ships seem to have made a great impression in the port's collective memory. Nonetheless, no fewer than three vessels flying foreign flags did tie up at Drogheda between 10 and 14 May 1847, some five months after Abdülmecid would have read the appeal issued on behalf of Queen Victoria for aid to be sent to Ireland.

The evidence here rests on the fine detail of a clipping which you can see on my website. According to this, one of these ships was probably Prussian: she was named the Meta, and her home port was Stettin, on the Baltic Sea. The other two, however, came from the Balkans, which were then still under Ottoman control. They were the Porcupine and the Ann – two rather English names for ships – and they apparently had English masters, for their names were Cleveland and Cloid (Floyd?) Nonetheless, they had sailed from Ottoman Selânik (today Thessaloniki, in modern Greece). All three merchantmen carried cargoes of wheat and "Indian corn" – cheap maize, one of Ireland's principal imports during the famine years.

This is a remarkable find – more so given the rarity of foreign commerce in small Irish ports in those days – but it is only fair to point out that the newspapers contain a strong indication that the cargoes of the Porcupine and the Ann were trade goods, not charity; they were consigned to local merchants based in Louth and Meath, who it might be expected would sell them rather than give them away. Speculation in imported corn, moreover, was scarcely unheard-of two years into the Great Famine; the Ottomans grew it, and the owners of the Porcupine and the Ann presumably knew that they did. It is, of course, still possible that the two ships were chartered on Abdülmecid's orders, and perhaps the only way of now guessing if they were would be to establish if they were the only vessels out of Balkan ports to unload cargoes in Irish ones during this period.

All in all it is possible to speculate that a remarkable chain of happenstance was forged in Drogheda during the starvation years. A wealthy queen's donation, a Sultan's noble gesture, and the coincidental arrival of unknown ships from unknown ports could well have supplied the germs of legends that continue to resonate in Ireland today.