‘ And the women said to Naomi, “Blessed be YHWH, who has not left you this day without a near kinsman, and let his name be famous in Israel.”
As a consequence of the birth of a son the women were able to say to Naomi, ‘blessed be YHWH Who has not left you without a near kinsman'. They were referring to the new born baby who would grow up to be head of Naomi's family, and would inherit the family property. Whilst Boaz would take her into his house (she becomes nurse to the baby) she was not strictly of his family. But the new born baby was in the eyes of the Law her son's son (in consequence of the law of Levirate marriage) and inherited the family land. Once he was of age he would thus have family responsibility for her.
“Let his name be famous in Israel.” Literally ‘and let his name be named in Israel'. The idea was that they hoped that he would ‘make a name for himself' by his success and godliness, so that he would be a successful family head.