The Midrash says that the husbands of the two women were father and son, making the two women, mother-in-law and daughter-in-law.
The two women had just lost their husbands, and needed a live child to exempt them from the status of a Yevamah. Both women gave birth to babies. However, these two babies were still less than 30 days old at the time that one of them died. The mother of the dead child would therefore be subject to the laws of Yibbum. This was the lying mother’s motivation for taking the other woman’s child.
If it were the mother-in-law’s child who had died, she would have no incentive to kidnap her daughter-in-law’s child. Even though her son (the deceased husband of her daughter-in-law) had passed away before her own husband had, and therefore he would not exempt her from Yibbum, nevertheless, she would be exempt from Yibbum for another reason. The living child was her son’s child, and a grandchild exempts one from Yibbum.
Only the daughter-in-law had the motive to lie and try to claim that the child was hers. If it was her baby who had died within 30 days of its birth, leaving her childless, she would have been bound to her husband’s brother as a Yevama- and that brother would have been -none other than the living baby (who was in fact her mother-in-law’s child – i.e., her deceased husband’s bother). Since her brother-in-law was a newborn, the daughter-in-law would have had to wait 13 years before this baby would be able to perform Chalitzah on her and free her to remarry.
King Solomon realized all of this and suspected that since the only one with a strong motive to lie was the daughter-in-law, the child must really belong to the mother-in-law.