A traditional Irish Stew would have mutton or lamb instead of beef because sheep are more commonly raised. Mutton would be the choice because for stew it had more flavor. Often the neckbones and shanks were put in to provide more flavor.
In this very cold country hearty root vegetables like parsnips, carrots and turnips were grown as the could be stored in a cellar all winter. This would be a given to put in their Irish stew.
Barley is a common grain in Ireland so that is often added.
Here is what Alan Davidson my favorite food historian has to say,
In Ireland many say that you should use "Neck Mutton Chops".....
"Irish stew is a celebrated Irish dish, yet its composition is a matter of dispute. Purists maintain that the only acceptable and traditional ingredients are neck mutton chops or kid, potatoes, onions, and water. Other would add such items as carrots, turnips, and pearl barley; but the purists maintain they spoil the true flavour of the dish. The ingredients are boiled and simmered slowly for up to two hours. Mutton was the dominant ingredient because the economic importance of sheep lay in their wool and milk production, and this ensured that only old or economically non-viable animals ended up in the cooking pot, where they needed hours of slow boiling. Irish stew is the product of a culinary tradition that relied almost exclusively on cooking over an open fire. It seems that Irish stew was recognized as early as about 1800..."
--- Oxford Companion to Food , Alan Davidson [Oxford University Press:Oxford] 1999 (p. 407)
An old Irish Stew Recipe
Take from two or three pounds of chops from the best end of a neck of mutton, and pare away nearly all the fat, for an Irish Stew should not be greasy. If liked a portion of the breast may be cut into squares and used, but a neck of mutton is the best joint for the purpose. Take as many potatoes as amount after peeling to twice the weight of the meat. Slice them, and slice also eight large onions. Put a layer of mixed potatoes and onions at the bottom of a stewpan. Place the meat on this and season it plentifully with pepper and slightly with salt. Pack the ingredients closely, and cover the meat with another layer of potato and onion. Pour in as much water or stock as will moisten the topmost layer, cover the stewpan tightly, and let its contents simmer gently for three hours. Be careful not to remove the lid, as this will let out the flavour."
--- Cassell's Dictionary of Cookery with Numerous Illustrations [Cassell, Petter, Galpin & Co.:London] 1874 (p. 331)
This is an example of the traditional Irish Pots that are used in Old Traditional Irish Cooking.
This is an example of a fireplace where Irish cooking took place with swinging arm pot holders.