I think it is probably an error that Elizabeth Lawrence (born 1674, Groton, Massachusetts, daughter of Nathaniel) married Joseph Gilson, because:
(1) Most importantly, on the same page as her birth record, there is an Elizabeth Lawrence who died at Groton in 1675 at age one (that is probably her) (see weblink below);
(2) There is another Elizabeth in the family who married Abner Harris, and Elizabeth's father Nathaniel lists daughter Elizabeth Harris (not Elizabeth Gilson) in his will.
(3) Joseph Gilson's wife is sometimes called Hepzibah rather than Elizabeth.
Therefore, I think Elizabeth (Lawrence) Harris had a sister Hepzibah (Lawrence) Gilson who also lacks a birth record. And that Elizabeth Lawrence born in 1674 died in 1675 and then the younger sister Elizabeth (Lawrence) Harris was named for her deceased older sister. I think this might have been missed because the 1675 records for Groton appear above (not below) those for 1674.