The question of "What is Toner's bog?" is a very interesting one precisely because there doesn't seem to be any really definitive answer.
My grandfather could cut more turf in a day
Than any other man on Toner's bog.
"Toner" is an old Gaelic name with roots in a Norse personal name. So first of all, Heaney is celebrating an ancient connection to Irish lineage by employing an ancient Gaelic name. Also, bogs have been historically used in Ireland for fuel; the turf (below the sod) is dug up and burned in the fire. This tradition continues today in rural areas. So secondly, there is the indication that Heaney is focusing attention on ancient Irish customs that relate to the sustenance of life. This would parallel the life sustaining work of gardening in "potato drills" "To scatter new potatoes that we picked." In summary, Toner's bog is a bog owned by or--very possibly--name for the Toner family of ancient lineage where Irish culture is revealed at a basic level and juxtaposed to the contrasting military conflict in Ireland through the word "gun” in the opening lines:
Between my finger and my thumb
The squat pen rests; as snug as a gun.
There is also a legendary atmosphere to the speaker’s description of his grandfather’s prowess with the spade on Toner’s bog. He cuts “more turf … / Than any other man”; he nicks and slices “neatly, heaving sods / Over his shoulder” to find the “good turfs” buried down beneath the surface sod. This may mark the grandfather as a heroic Irish cultural figure who is meant to be a symbol of the qualities of all Irish people (especially as he drinks milk instead of whiskey, thus breaking the stereotype).
Toner's bog in the poem "Digging" refers to the owner of the particular plot of bogland. In Ireland, in the absence of coal, peat taken fromthe bog is used for turf fires as heat. The cutting of turf is labour intensive, and during the time Heaney speaks of would have been cut by hand. A person would pay the owner of the bog for the turf he would cut for his own use.