It appears from the source below that Elishua b. 1637 was a daughter, and Elisha b. 1641 was a son. The records from First Church are likely Baptismal records and not birth records. Elisha being bapt. aged about 13 days.
Other sources indicate dau. Elishua - died young but it is not recorded here.
SOURCE: City Document 130: Boston Births. Baptisms, Marriages and Deaths: 1630-1699: Genealogical Pub. Co. 1978:
(p. 5): First Church
1637: Elishua daughtr of Edward Hutchinson the younger 5 day: 3 Mo.
(p. 12) First Church
1641: Elisha of Edward Hutchinson the younger , aged about 13 days 28 day 9 mo
The father, Edward Hutchinson would be PGM.