The Midrash (Koheles Rabah 10:16) tells us that the reason both of these women were so desperate to have the living child declared theirs was that they were both potential Yevamos (widows subject to Yibbum). Neither of the two had any other offspring. Whoever would be judged to be the childless woman would not only lose the infant, but would also be trapped in the unpleasant status of Yevamah, being dependent upon her brother-in-law’s good will.
The Midrash (Yalkut Shimoni 2:175) asserts that the husbands of the two women were father and son, making the two women, mother-in-law and daughter-in-law to each other.
According to the Meiri in his commentary to Yevamos 17a, the two Midrashim may be complementing each other.