Not sure if this thread is still of interest, but it involves several surnames and villages I've researched, so it intrigued me enough to dig through my notes and the other sources I have to hand.
The 3 Sep 1839 marriage of Henry White and Sarah Stoodley tells us she was a minor, and of Hythe, the daughter of John Stoodley, a miller. She seems to have been buried on 20 Nov 1848 at Hythe, aged 28, which puts her birth in 1820, which is not quite consistent with her being a minor when she married, nor with her being 20 at on 6 Jun 1841 when the census was taken, but it's not far out. The baptism at Beaulieu on 1 Apr 1821, daughter of John and Elizabeth Studley, which was posted above seems almost certainly the right one.
John and Elizabeth Studley / Stoodley had other chidlren baptised at Beaulieu: Ann (24 Mar 1816), Mary (3 Aug 1817), George (7 Feb 1819) and John (12 Apr 1823). Ann was spelt Studley, the rest Stoodley. Their 12 Oct 1814 marriage at Beaulieu has already been pointed out which gives Elizabeth's previous surname as Rose. I cannot find any earlier instances of the surname in the Beaulieu parish register, nor any burial for them, nor any reference to them on the census. The surname is common enough a few miles along the coast in Lymington, but without an age or some concrete clue that John was really from Lymington, it's hard to be sure. We have much the same problem with Elizabeth Stoodley née Rose. There are plenty of plausible baptisms but none in Beaulieu, and no way of being sure which is the right one.
There's a baptism on 14 Oct 1808 at Fawley of a Henry White, son of James White and Jemima née Hempson. The mother is almost certainly the Jemima White of Hythe who was buried 6 Dec 1848 at Fawley, aged 69, putting her birth in about 1779. This is consistent with the woman on the 1841 census with Henry and Sarah. There are several James Whites in the Hythe area at the time, but if we assume James was within a decade of Jemima in age the only candidate burial is the James White of Hythe buried on 30 Sep 1827 at Fawley, aged 51, which would put his birth in about 1776.
James and Jemima White's children were bapised variously in Fawley or Dibden between 1802 and 1821: Sarah (9 Jun 1802 at Fawley and 4 Jul 1802 at Dibden, the latter described as being of Fawley), Jemima (18 Sep 1803 at Fawley), William (22 Jun 1806 at Dibden, described as being of Hythe), Henry (14 Oct 1808 at Fawley), Anne (26 Dec 1813 at Dibden), Charles (24 Mar 1816 at Dibden), Josiah (7 Jun 1818 at Dibden) and Thomasin (6 May 1821 at Dibden). The mixture of parishes makes sense. Hythe was at this time a detached part of Fawley parish, surrounded by Dibden parish. There's a marriage for James White junior and Jemima Hemson on 25 Dec 1800 at Fawley, described as a bachelor and spinster, respectively, of Hythe. It was witnessed by James White senior and Ann Bider.
There's a 12 Oct 1775 private baptism at Fawley for a James son of James and Sarah White, followed by a record of the son being received into the church on 26 Dec 1776 at which the parents are noted as being of Hythe. This fits both the age of the burial, the fact that James and Jemima lived in Hythe, and the use of 'junior' in describing James when he married. However it is not the only option. There is also a 23 Dec 1783 baptism at Fawley for a James son of James and Rebeckah White of Hardly [sic], which would also explain the use of 'junior' and does not make him unreasonably young to have married in 1800. It's certainly possible that it was this child who later married Jemima, while the 1775 child died in 1827. It is would mean he moved from Hardley to Hythe, but they're only 2½ miles apart. The 1775 baptism is seems the more probable, but you'd need additional evidence before you could be sure.
Jemima Hempson [or Hemson] is a very unusual name. The surname as spelt is not a Hampshire surname and I cannot find a single instance of it anywhere in the parish registers for the whole of mainland Hampshire before the 19th century. It could be a misspelling or mistranscription of Hanson or Hampson or similar, which are slightly more common, but there are no recorded Jemimas with these surnames in the county. Further afield, there's a Jemima Cally Hempson, daughter of Thomas and Mary Hempson, born 13 Dec 1780 and baptised 5 Jan 1785 in Kenton, Devon, a few miles south of Exeter. But you'd want more than a matching name before jumping to that conclusion, especially as Jemima is listed as being born in the county on the 1841 census and of Hythe at the time of her marriage.
That's as far as I can get at the moment, and I'm afraid it hasn't linked to any of the families I'm researching.
