The source of all these claims is an article on website of BibleBelievers that talks about 'Giant Humans and Dinosaurs', which says that the original source of of it is an article from December 1895 issue of The British Strand Magazine that was reprinted in a book called "Traces of the Elder Faiths of Ireland" written by W.G. Wood-Martin.
"Pre-eminent among the most extraordinary articles ever held by a railway company is the fossilized Irish giant, which is at this moment lying at the London and North-Western Railway Company's Broad street goods depot, and a photograph of which is reproduced here. . . This monstrous figure is reputed to have been dug up by a Mr. Dyer whilst prospecting for iron ore in County Antrim.
The principal measurements are: entire length, 12ft. 2in.; girth of chest, 6ft. 6in.; and length of arms, 4ft. 6in. There are six toes on the right foot. The gross weight is 2 tons 15cwt.; so that it took half a dozen men and a powerful crane to place this article of lost property in position for the Strand magazine artist. Dyer, after showing the giant in Dublin, came to England with his queer find and exhibited it in Liverpool and Manchester at sixpence, sixpence a head, attracting scientific men as well as gaping sightseers".
THE ACTUAL FACTS
W.G. Wood-Martin, who shared the original source of this article from The Strand Magazine did not actually believe in the claim and share it. He mentioned the story as printed in the magazine and then said - "let the reader judge as to the genuineness of the fossilized Irish giant" (You can see the picture from the book in image section below). The original article in Strand Magazine, 'Lost Property Office' written by William G. FitzGerald briefly mentioned the same story, followed by an ownership dispute between the two showmen.
It is mentioned that Mr. Dyer started a show business with the Giant, exhibiting it in Dublin and then in England. Later he induced a showman named Kershaw and sent this Giant from Manchester to London by rail. This is where the picture in question was taken. However, due to an ownership dispute and that the rail carriage charge was not paid, the Giant was left with the rail company and nothing was heard about it ever since.
There are few aspects that question the authenticity of this fossilized Irish Giant:
The Strand Magazine was primarily a fiction magazine, which sometimes used to publish puzzles and brain-teasers as well.
The only available proof of the story is the picture, which again is not sure to be precise. Couldn't it be a carefully selected camera angle shot with the freight car in the background so as to make it appear Giant size?.
If the story was a fact and it became popular, then why there are no scientific examinations of this particular "fossil" before it was "lost."
About the claimed weight of the Giant, 2 British tons and 15 cwt (Centrum weight) is equivalent to 6,160 pounds (considering 1 British ton equal to 2240 pounds), which is extraordinary.