Sources disagree on his migration date (the one in red is wrong):
Migration:  1637, arrived Ludlow, MA, on the Confidence with "brothers" Edmund and William
Migration:  27 Apr?/Jun? 1638, arrived Boston, MA, with Thomas LECHFORD party

Father possibly:  Thomas? BROWN
Mother said to be:  Sarah __?__

It is said that Sarah died in childbirth and that Thomas was adopted by his uncle, Edmund BROWNE.  But why would this be in doubt one way or another?  It appears he was baptized, and that record should tell us the name of his father and, possibly, at least his mother's given name.

Bridget's father is said by some to be the William BATEMAN whose will was proved 10 Oct 1658 in Fairfield, CT, but that will does not mention a daughter, Bridget.  Also, while we don't know where William BATEMAN was born, we do know that one of his children was born in Toppesfield, Essex, so it's unlikely he's the father of children born in Hawkedon, Suffolk.

Has someone mis-read 1651 as 1681?

How do we know she was born in Hawkedon?  The distance from Lavenham to Hawkedon, as the crow flies, is about 7 miles, and about 10 miles by road.

1.  Sir John Bernard Burke.  1996.  "Burke's American Families with British Ancestry." Pages 2529-3022 from Burke's Genealogical and Heraldic History of the Landed Gentry (1939, Burke's Peerage, London).  Genealogical Publ. Co., Baltimore, MD.  On p. 2584:

BROWN formerly OF LAVENHAM... THOMAS BROWNE, of Lavenham, Suffolk, and afterwards of Sudbury, Massachusetts, whither he went with his brother, Edmund [in 1638], bapt. at the Church of St. Peter and Paul, Lavenham, 10 Jan 1605, and d. 3 Nov. 1688.  His son,  ELEAZER BROWN...

2.  Clarence Almon Torry.  1985.  New England Marriages Prior to 1700.  New England Historic Genealogical Society, Boston, MA (reprinted 1997, Genealogical Publ. Co., Baltimore, MD):

The "b" indicates the birth date of a child as an indication that the marriage took place before this time.  The locations are known residences of the husband, ending with the husband's place of death.  An entry in Torrey does not imply a marriage in New England; it only demonstrates the presence of a married couple there before 1700.  Torrey is a compilation of published sources, not primary records, so he should be considered a secondary source.

p. 109

BROWN, Thomas (-1688) & Bridget __?__; b 1642; Concord/Cambridge... BROWN, Thomas (-1690, 1691) & Martha (EATON) OLDHAM (1630-), w Richard; 7 Oct 1656; Cambridge

3.  Charles Henry Pope.  1900.  The Pioneers of Massachusetts. C.H. Pope, Boston, MA (republ. 1965/98 by Genealogy Publ. Co., Baltimore, MD; Broderbund CD-504; online at Ancestry.com; Kindle e-copy sucks; paperback copy from NEHGS):

p. 75

Thomas, Sudbury, propr. 1640.  The Gen. Court 7 (8) 1640, granted him 200 acres of land for the 25 li. adventure of Mrs. Anne Harvyes.  He seems to have rem. to Concord, where by wife Bridget he had ch. Boaz b. 14 (12) 1641, Mary b. 26 (1) 1646, Eliezer b. 5 (3) 1649.  He is clearly the man who was adm. chh. Cambridge May 18, 1666; had m. before 1658 Martha, widow of Thomas Oldham; ch. Mehitabel bapt. June 2, 1661, Mary bapt. March 8, 1663, Ebenezer bapt. July 23, 1665, Ichabod bapt. Sept. 9, 1666. [Mi.] Will dated Nov. 23, 1690, prob. in Jan. folg., beq. to wife Martha, daus. Mehitabel and Martha, and sons Ebenezer and Ichabod. The inv. shows house and land in Camb. and 300 acres at Worcester.

4.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com):  there is a file here that appears very well documented.  I'm going to run with it and try to fill in the documentation.