Very Early Oakmans in Co Antrim
The Widow Oakeman
The Hearth Money Rolls of 1669 lists Widow Oakeman living in either Ballymoneymore, Ballyminimore, Tullynewbane or Tullynewbank Townlands - her home contained one hearth which meant Mrs Oakeman was liable to pay a tax.
Next mention of an Oakman in the Glenavy area is in 1705 on an old headstone in Glenavy Parish Church burial ground. This may well indicate the death of the Widow Oakeman and likely where her husband was buried sometime prior to 1669.
Robert Oakman, Glenavy Parish
In 1709, Robert Oakman is listed as a Surveyor (Glenavy upper part) for Glenavy Parish Church (see Glenavy Past and Present by Charles Watson). Could Robert have been a son of the Widow Oakeman?
Robert Oakman's Will (extract only - PRONI T/700 p372), dated 1725 and proven in 1729, tells us he was married to Eliza and had four adult children living at the time he made his Will. They were Walter, William, Elenor and Margaret. In the records of Glenavy Parish Church, we find Mrs Elizabeth Oakman marrying William Thompson in 1732. This is likely to be Robert's widow remarrying a few years after Robert died.
From this we have the following family tree:

Oakman Family Tree to 1725
Robert's Children:
Walter Oakman, Ballydonaghy
The earliest we find Walter Oakman is in the Vestry Records of Glenavy Parish Church where he is "present at Vestry" in 1714 and 1715. Walter would have to have been 21 years or older to attend vestry meetings, making the date of his birth sometime before 1693. Some years later, in 1729, we find Walter leasing land in the Townland of Ballydonaghy, which is situated in the Parish of Camlin, just north of Glenavy Parish. Walter's name appears on two pieces of land, each adjacent to the other - one with someone named Hurdman or Herdman and the other with someone named Johnston (PRONI D427 pages 220 & 61-2).
These joint leases might be providing us with the names of others connected with the Oakman family. In the 1725 Will Extract for Robert Oakman (PRONI T/700 p372), we find that one of Robert's daughters, Elenor, had married a man named Herdman. It is possible that Walter Oakman was leasing land with his brother-in-law. The name Johnston might be the family name of Walter's wife. More research will need to be undertaken before these assumptions can be proven.
Walter was a Linen Draper (a dealer in linen). This occupation likely involved a shop where the linen could be sold and may have involved travelling to other cities and towns. Walter probably had premises in Belfast. In those days linen was woven from the fibres of flax plants and was very labour intensive.
Walter died about 1755. An extract from his Will (PRONI T/700 p373), proven in May 1755, mentions children John, Mary, Elizabeth, Walter, Nicholas, Robert, Catherine, Jean and Isabella. Walter also made mention of his brother, William, who appears to have died sometime after Walter's death in 1755.
From the Will extract of Walter we now have the following family tree:

Family Tree of Walter Oakman
William Oakman, Aghadalgon
In 1729 we also find Walter's brother, William, leasing land in the Parish of Glenavy in the Townland of Aghadalgon (PRONI D427 pages 220 & 61-2). William's name appears alongside the name Lennon or Lennox.
We also find William "present at Vestry" meetings at Glenavy Parish Church in 1728, 1733 and 1745.
Elenor Oakman
Elenor married a man named Herdman or Hurdman and had at least one child - a son, John Herdman, who was born before 1725.
Margaret Oakman
Margaret married a man named Ker sometime before 1725.