What happens to Ruth after the birth of Obed? Not much is known, but Avivah Zornberg discusses how the the rabbis try mightily to write Ruth back into the text through midrash, originating with a verse in I Chronicles.
הֵ֚מָּה הַיּ֣וֹצְרִ֔ים וְיֹשְׁבֵ֥י נְטָעִ֖ים וּגְדֵרָ֑ה עִם־הַמֶּ֥לֶךְ בִּמְלַאכְתּ֖וֹ יָ֥שְׁבוּ שָֽׁם׃
These were the potters who dwelt at Netaim and Gederah; they dwelt there in the king’s service.
(ב)... הֵמָּה הַיּוֹצְרִים, זֶה בֹּעַז וְרוּת. וְישְׁבֵי נְטָעִים, זֶה שְׁלֹמֹה שֶׁהָיָה דוֹמֶה כְּנֶטַע בְּמַלְכוּתוֹ. וּגְדֵרָה, אֵלּוּ סַנְהֶדְּרִין שֶׁהָיוּ גּוֹדְרִים עִמּוֹ בְּדִבְרֵי תוֹרָה. עִם הַמֶּלֶךְ בִּמְלַאכְתּוֹ יָשְׁבוּ שָׁם, מִכָּאן אָמְרוּ לֹא מֵתָה רוּת הַמּוֹאֲבִיָּה עַד שֶׁרָאֲתָה שְׁלֹמֹה בֶּן בְּנָהּ יוֹשֵׁב וְדָן דִּינָן שֶׁל זוֹנוֹת, הֲדָא הוּא דִכְתִיב (מלכים א ב, יט): וַיָּשֶׂם כִּסֵּא לְאֵם הַמֶּלֶךְ זוֹ בַּת שֶׁבַע, וַתֵּשֶׁב לִימִינוֹ זוֹ רוּת הַמּוֹאֲבִיָּה.
... 'These were the potters' refers to Ruth and Boaz. 'And those that dwelt among plantations' refers to Solomon who was like a plant in his kingship. 'And hedges': these are the Sanhedrin who with him made a hedge round the words of the Torah. 'There they dwelt, occupied in the king's work.' On the strength of this verse they said that Ruth the Moabitess did not die until she saw her descendant Solomon sitting and judging the case of the harlots. That is the meaning of the verse, And caused a throne to be set for the king's mother, i.e. Bath Sheba, And she sat at his right hand (I Kings 2:19), referring to Ruth the Moabitess.
וַתָּבֹ֤א בַת־שֶׁ֙בַע֙ אֶל־הַמֶּ֣לֶךְ שְׁלֹמֹ֔ה לְדַבֶּר־ל֖וֹ עַל־אֲדֹנִיָּ֑הוּ וַיָּקָם֩ הַמֶּ֨לֶךְ לִקְרָאתָ֜הּ וַיִּשְׁתַּ֣חוּ לָ֗הּ וַיֵּ֙שֶׁב֙ עַל־כִּסְא֔וֹ וַיָּ֤שֶׂם כִּסֵּא֙ לְאֵ֣ם הַמֶּ֔לֶךְ וַתֵּ֖שֶׁב לִֽימִינֽוֹ׃
So Bathsheba went to King Solomon to speak to him about Adonijah. The king rose to greet her and bowed down to her. He sat on his throne; and caused a seat to be set for the king's mother; and she sat on his right hand.
The famous story of the two harlots coming to King Solomon to claim the the one surviving baby of the two children they have borne is found just after this in I Kings 3:16-28. This case and Solomon's judgment is depicted as evidence of his great wisdom and impartiality as a judge.
Zornberg discusses Ruth's reappearance here with Solomon in two contexts:
1) Ruth has been at least figuratively kept alive, in the rabbis' world, in order to see the future result of the hesed she performed in her lifetime. Yes, she redeemed her husband's name and Naomi's husband's name, but she was also the maternal ancestor of not only King David, but his children as well.
2) Zornberg connects the birth and "loss" of her child Obed to the specific case of the living and dead babies before King Solomon, and also discusses the verse in Kings in the context of a choice that Solomon may have made in deciding who his real mother is, similar to Ruth, who once upon a time chose for herself a new mother.
Source: "Law and Narrative in the Book of Ruth," The Murmuring Deep, pp. 375-379.
וַיִּשְׁמְע֣וּ כָל־יִשְׂרָאֵ֗ל אֶת־הַמִּשְׁפָּט֙ אֲשֶׁ֣ר שָׁפַ֣ט הַמֶּ֔לֶךְ וַיִּֽרְא֖וּ מִפְּנֵ֣י הַמֶּ֑לֶךְ כִּ֣י רָא֔וּ כִּֽי־חָכְמַ֧ת אֱלֹקִ֛ים בְּקִרְבּ֖וֹ לַעֲשׂ֥וֹת מִשְׁפָּֽט׃
When all Israel heard the decision that the king had rendered, they stood in awe of the king; for they saw that he possessed divine wisdom to execute justice.
ויהי בימי שפט השפטים Now it was in the days that the judges were judging. (1:1)
We have learned that this was a period in Israel's history in which leadership, which was in the hands of judges, was corrupt.
Is it not fitting that the rabbis bring back Ruth to see the return of justice and wisdom in the judgements of King Solomon? And for her to know that she had a role in the history leading up to this?