Statutory birth registration only started in Ireland in 1864 so there are no births in those records prior to that year. You may find childrens baptisms in church records but they are held separately, in PRONI, and not all are on-line.
I see John Shaw in Balleny in Griffiths in 1863. He had plot 92 which was a house and 1 and a half acres. It remained in his name until 1878, when he was succeeded by Robert Craig. Possibly he died around that time? Griffiths lists several Martin families in Ednego. No Henry so possibly he was dead or had moved away.
Much of the population in Ulster were involved in weaving.
The average weaver in Ireland in the 1700s and first half of the 1800s was a labourer who often lived in a small cottage on someone else’s farm, and would have a few perches of land to grow some vegetables and some flax (the raw material for linen). Rent would generally be paid by an agreed number of days labour on the farm, though occasionally it was paid in cash. The labourer was otherwise free to undertake any additional work that might be available, possibly on another farm or on government schemes such as road improvements.
Weaving was top up income for small farmers and labourers in Ulster. It was originally undertaken at home using hand powered looms (such as are still in use in the Outer Hebrides for making Harris Tweed). The looms were portable and could be packed up when not needed or when moving house. Men and women both did the weaving, with women and children spinning thread and other related work. It provided a bit of extra income, and cash (in a society that was largely run on the barter system). As well as purchasing the things that barter can’t buy eg a ticket to Canada or America, the cash ensured that their lot was slightly better than for people in other parts of Ireland where linen was not made. (Most Irish linen was made in Ulster).
As the 18th century passed, new inventions led to the building of mechanized weaving factories, powered by water or steam. These could make linen far more quickly and to a higher standard than most hand loom weavers could achieve and so home weaving went into decline.
The mechanized linen mills not only put paid to home weaving it particularly affected men. In the mills it was mostly women and children who tended the machines (cheaper and with more nimble fingers). There were very few men in the mills, just the odd engineer and some overseers. So a whole tranche of men lost their weaving work. (Causing many to look for work outside Ireland). On the other hand it brought huge social change and improvement for women who, for the first time, could gain some independence and earn a proper wage instead of having to work around their parents cottage or farm, 7 days a week, for their board and lodging.
Most of the mechanized mills were in towns such as Belfast, Lisburn and Ballymena. By around 1850 Belfast was known as Linenopolis because of all the linen mills there. By 1870 it was the largest linen producing area in the world. Weaving history on this link:
Here’s a link which will tell you a bit about the famine in Co Down:
The Andrews family of Comber are mentioned in the above article. They were a famous local family, involved in shipbuilding, politics, railways, the law and milling. Thomas Andrews was the builder of the Titanic (and went down with the ship).