The children of William and Ruth (Hunt) Secord
William's will names his daughters, though unfortunately not with their married names, as Rachel, Anne, Sarah, Elizabeth, Olive, Susan and Mary, though Anne had predeceased him and her share would go to her heirs. The will also mentions William's son William who had also predeceased him. The other sons were not listed, though he indicated that there were four of them, and then named his son James as executor. The remaining sons are said to be Gilead, Elijah, and Carmel. I think this is correct as the family of Elias Secord, head of the other early Secord family in New Brunswick, seems to be documented from perhaps a family bible record, as there are names and complete dates of birth for most of them in a number of places on the Internet, and Gilead, Elijah, and Carmel are not among them. Also, Gilead, Elijah, and Carmel seem to be associated with Kings County, where William lived, and not Queens County, where Elias lived.
I think that the daughters were listed in order of birth in the will, because Anne was listed in second place though she was the only one who had died before her father, and also because that seems to fit with what I can determine of their approximate dates of birth. We know William Junior's date of birth from his baptism record. (In the registers for the First and Second Presbyterian Church of New York City, there is a baptism record for William Secord on April 16, 1775. His date of birth was given as Feb. 18, 1775.) The other sons, like many of the daughters, can be found in census records which give us an estimate of dates of birth.
Here are the sons and daughters, listed in approximate order, and with estimated birthdates for all but William:
1) William Secord the younger - born Feb. 18, 1775
2) Rachel Secord - born about 1776-1777
3) Anne Secord - born about 1778-1781
4) Sarah Secord - born about 1782-1783
5) Elizabeth Secord - born about 1782-1783
6) Olive/Olivet Secord - born about 1786
7) Gilead Secord - born about 1787-1788
8) Elijah Secord - born about 1790
9) Susan Secord - born about 1792-1795
10) James Secord - born about 1795-1796
11) Mary Secord - born probably 1798
12) Carmel Secord - born about 1803
The records seem to indicate that the first five were born in New York, which fits with their arrival in what is now New Brunswick in the summer of 1783. (See http://www.carensecord.ca/locations/NewBrunswick/Secord/WandESecordPassagetoNB.html for more information.)
Ruth (Hunt) Secord was born about 1757 or 1758. (See Ruth (Hunt) Secord.) If her first child was born in 1775 and the last about 1803, then she was about 17 for her first child and about 45 for her last one. Of course there may also have been other children who died young, but it seems unlikely that she had others before 1775 or after 1803.
The names Olivet, Gilead and Carmel are of special interest because they are unusual. I don't see them in any Secord families to that point. It is possible that two of them came from geographic names of New York state, as there is a Carmel there, and also a Lake Gilead, both in Putnam County. However, there is also at least one Gilead Hunt in NY during the time period of interest here, and it seems a possibility that he may have been Ruth's relative, for whom her child was named. I have seen a transcription of a will for a Moses Hunt on the Ancestry site. He was of East Chester in Westchester County, and the will was dated Jan. 30, 1760 and proved May 21, 1764. Evidently his wife had predeceased him, but his children were listed as Benjamin, Timothy, "Gillead", Reuben, Mary Yeamans, Sarah Oakley, Martha Tippett, Eunice Oakley, Phoebe Oakley, and Rebecca Gee. Clearly, Ruth was not a child of this Moses Hunt as he was too old to have a two-year-old daughter when he wrote this will, as all of his other daughters were already married. Also, this Moses was not from Newboro, which I have assumed to be Newburgh, in Orange County near Ulster County. There were likely other related Hunt families in the area, not documented online as yet, and they may carry some of the same first names, including possibly Moses and Gilead, and Ruth may come from one of those families. It is also possible that the names Olivet, Gilead and Carmel were simply chosen for religious reasons, as they are all biblical.
Rachel and Anne were likely names with some significance for William and Ruth, assuming that they were the first two daughters. Were those the names of their mothers, perhaps?