There is a direct parallel to the Book of Genesis that provides us a clue.
In the Book of Genesis Rachel provided to Jacob her maidservant, Bilhah, so that Bilhah could bear proxy children on behalf of Rachel.
Genesis 30:3-4 (NASB)
3 She said, “Here is my maid Bilhah, go in to her that she may bear on my knees, that through her I too may have children.” 4 So she gave him her maid Bilhah as a wife, and Jacob went in to her.
So the baby Obed was placed on the lap of Naomi according to Ruth 4:16, whereupon Naomi "became his nurse."
The imagery here is that Naomi had restored to her the Promised Seed, because we see that "the elders of the court" mention Tamar, and so they place Naomi in the same context as Tamar...
Ruth 4:12 (NASB)
12 Moreover, may your house be like the house of Perez whom Tamar bore to Judah, through the offspring which the Lord will give you by this young woman.”
In other words, the "elders of the court" had recognized that Naomi had "captured" the Promised Seed in the same way that Tamar "captured" the Promised Seed from Judah, for whom the Levirate marriage was unfulfilled. (Judah later indicated that Tamar was more righteous than he, because she recognized, and therefore embraced, the Abrahamic Covenant of the Promised Seed through faith -- please see Genesis 38:26.) Naomi "captured" the Promised Seed of Abraham because (like Tamar) she cunningly orchestrated the consummation of the Levirate promise -- that is, she brought Boaz and Ruth together through meticulous planning and execution (like Tamar), since their sights were on this Promised Seed of Abraham through the fulfillment of Levirate marriage.
Please remember that Naomi's husband was Elimelech ("my God is king"). Elimelech was descended from Judah, which had the promise of the scepter, or rulership among the tribes of Israel (Genesis 49:10). That is, the grandfather of Boaz was the SAME grandfather of Elimelech. This name of this grandfather was Nahshon, who was the leader of the tribe of Judah (1 Chr 2:10). Thus Elimelech (and now Boaz) continued this royal line of the tribe of Judah. Like Tamar, Naomi recognized the promise of the Promised Seed (through the tribe of Judah), and accordingly acted on her faith.
Thus Naomi snatched victory from the jaws of defeat, because by faith she recognized the Promised Seed, who (unbeknownst to her) was later going to be King David (who also would receive a promise of HIS descendant, who would be the ultimate Promised Seed to sit on the "eternal" throne of Israel). This Promised Seed is Ha Meschiach: kindly compare Micah 5:2 with Ruth 4:11 and note the direct correlation of Bethlehem/Ephrathah in both verses, and of course the direct reference to eternality ("from everlasting" or מִימֵי עֹולָם) in Micah 5:2.
In conclusion, Naomi is a "mother" to Obed because she orchestrated the consummation of the Levirate marriage, and therefore "captured" the Promised Seed of the tribe of Judah by faith (exactly like Tamar). Obed was therefore laid on Naomi's lap in the same way that Bilhah's offspring were laid on the knees of Rachel (as proxy children), and so Naomi was the "mother" of Obed (who became the father of Jesse, who became the father of King David, whose descendant is Ha Meschiach -- i.e., the Messiah)....
ADDENDUM (to answer comment, below):
What made Elimelech and his line so “special”? Elimelech and Naomi were part of the faithful remnant in Israel, and therefore they embraced the hope of the Promised Seed of Abraham, who was identified as an individual when God made several promises to Abraham in Genesis 22:17-18. (The possessive personal pronoun suffixes in Hebrew, which modify nouns in those verses, are in the masculine singular.) That is, through this seed (masculine singular) the nations of the earth would be blessed. Additionally, as mentioned above, before his death, Jacob indicated that the scepter would never depart the tribe of Judah, which suggested that the future rulers of the Israelites would be from the tribe of Judah. Finally, Moses predicts that a future prophet (or Second Moses) will rise up and lead the Israelites (Deut 18:15). So this hope (intertwined between these promises) was the basis of hope of a leader/savior in the Hebrew Bible. Faith on the Promised Seed (masculine singular), as evidenced by Tamar and Naomi, was the basis of this hope. In the collective sense, the seed of Abraham were the Israelites, but in the individual sense, the seed of Abraham was to be the savior (and leader) of Israel.
Now going back to the text of Scripture, we see that Amminidab (from the tribe of Judah) did not cross the Jordan River, and therefore his son Nahshon entered the Promised Land with Joshua, who in turn had appointed him (Nahshon) to be the head or leader of the tribe of Judah (Num 10:14 and 1 Chr 2:10). Nahshon’s son was Salma (or Salmon). Salma sired a son with Rahab the prostitute, whose name was Boaz (Matthew 1:5). Now Elimelech was indeed a “near relative” of Boaz, which meant that Nahshon was their mutual grandfather. (Whether or not Elimelech was related to Salma is unknown.)
As we just mentioned, Elimelech and Naomi were Israelites who were part of the faithful remnant, which means that they embraced the hope of the Promised Seed. But as circumstances would have it, her sons married Moabite women at which time Naomi’s husband and her two sons died. As a result Naomi almost lost her faith as well. (By the way Ruth the Moabitess became a part of this faithful remnant as a prosylyte to Judiaism.) When Ruth later met and married Boaz through Naomi’s careful planning, Boaz had supplanted Mahlon (compare Ruth 4:10 with Deut 25:6). Therefore Boaz’s genetic biological seed continued the ROYAL tribe of Judah and line of Elimelech-and-Mahlon notwithstanding that both Boaz (son of a prostitute) and Ruth (Moabitess) were biological racial outcasts (cf. Judges 11:1-2).
The point of the story is that the Promised Seed of God in the Hebrew Bible is not restricted or confined to genetics and biology, but also includes people of faith. People of faith in the Hebrew Bible are not just people biologically related to Abraham, but also people who embrace the Promised Seed by faith as outsiders (e.g., Tamar, Rahab, and Boaz-and-Ruth) and who therefore became biological progenitors of the Promised Seed. What made Elimelech and Naomi “special” therefore was that they were Israelites who embraced the hope of the Promised Seed. But later, alone with Ruth, Naomi then had acted on her faith "to capture" Boaz through Levirate marriage, and thus she was equated with Tamar by the elders of Bethlehem. Please note that not even Tamar is recorded as a biological Jew, but she nevertheless “captured” the hope of the Promise Seed from Judah, and therefore was declared a righteous woman (Genesis 38:26).