There are many sources that speak about Shaul Hamelech's righteousness.
To add to what has already been said, I will base the bulk of my answer from a Mishnas Rebbi Eliezer, but will also add other sources that support the points.
In Mishnas Rebbi Eliezer 10 (starts at the bottom of the page) it notes Shaul's overarching humility. At the time that Shmuel said to Shaul, "And for whom is all of Israel yearning" (Shmuel I 9:20), Shaul answered "But I am only a Benjaminite...". And at the time that they sought to anoint him as king, he went and hid himself (Shmuel I 10:22).
As a result it quotes Rav Yanai who notes 12 separate character attributes for which Shaul deserves praise:
- Humility - (as per his answer above about being "only" בֶן־יְמִינִ֤י אָ֙נֹכִי֙).
Also worth noting the Yalkut Shimoni, Shmuel 117 that says that he initially refused to accept the kingship, but when he heard that a chacham (a sage), a chassan (a bridegroom) and king are forgiven for all their previous sins he accepted.
- He listened to the reproach of others and remained silent - See Shmuel I 10:27
וּבְנֵ֧י בְלִיַּ֣עַל אָמְר֗וּ מַה־יֹּשִׁעֵ֙נוּ֙ זֶ֔ה וַיִּבְזֻ֕הוּ וְלֹֽא־הֵבִ֥יאוּ ל֖וֹ מִנְחָ֑ה וַיְהִ֖י כְּמַחֲרִֽישׁ׃
But some scoundrels said, “How can this fellow save us?” So they scorned him and brought him no gift. But he pretended not to mind.
Although the Yalkut Shimoni, Remez 117 points out - "אמר רבי יהודה אמר רב מפני מה נענש שאול מפני שמחל על כבודו שנאמר ובני בליעל אמרו מה יושיענו זה ויהי כמחריש" that it was this fact that led to his ultimate punishment.
- He forgave insults - See Shmuel I 11:12-13
וַיֹּ֤אמֶר הָעָם֙ אֶל־שְׁמוּאֵ֔ל מִ֣י הָאֹמֵ֔ר שָׁא֖וּל יִמְלֹ֣ךְ עָלֵ֑ינוּ תְּנ֥וּ הָאֲנָשִׁ֖ים וּנְמִיתֵֽם׃ וַיֹּ֣אמֶר שָׁא֔וּל לֹֽא־יוּמַ֥ת אִ֖ישׁ בַּיּ֣וֹם הַזֶּ֑ה כִּ֥י הַיּ֛וֹם עָשָֽׂה־יְהוָ֥ה תְּשׁוּעָ֖ה בְּיִשְׂרָאֵֽל
The people then said to Samuel, “Who was it said, ‘Shall Saul be king over us?’ Hand the men over and we will put them to death!” But Saul replied, “No man shall be put to death this day! For this day the LORD has brought victory to Israel.”
- He was cautious with the mitzvah of tzitzis - see Shmuel I 24:5
וַיָּ֣קָם דָּוִ֗ד וַיִּכְרֹ֛ת אֶת־כְּנַֽף־הַמְּעִ֥יל
David went and stealthily cut off the corner of Saul’s cloak.
- He cared for the poor daughters of Israel, in that all those who did not have dowries, he supplied them with -see Shmuel II 1:24
בְּנוֹת֙ יִשְׂרָאֵ֔ל אֶל־שָׁא֖וּל בְּכֶ֑ינָה הַמַּלְבִּֽשְׁכֶ֤ם שָׁנִי֙ עִם־עֲדָנִ֔ים הַֽמַּעֲלֶה֙ עֲדִ֣י זָהָ֔ב עַ֖ל לְבוּשְׁכֶֽן
Daughters of Israel, Weep over Saul, Who clothed you in crimson and finery, Who decked your robes with jewels of gold.
- He demonstrated the virtue of shame - See Shmuel I 15:25
'וְעַתָּ֕ה שָׂ֥א נָ֖א אֶת־חַטָּאתִ֑י וְשׁ֣וּב עִמִּ֔י וְאֶֽשְׁתַּחֲוֶ֖ה לה
Please, forgive my offense and come back with me, and I will bow low to the LORD.”
- He was tznua (modest) - See Shmuel I 24:4
וַ֠יָּבֹא אֶל־גִּדְר֨וֹת הַצֹּ֤אן עַל־הַדֶּ֙רֶךְ֙ וְשָׁ֣ם מְעָרָ֔ה וַיָּבֹ֥א שָׁא֖וּל לְהָסֵ֣ךְ אֶת־רַגְלָ֑יו וְדָוִד֙ וַאֲנָשָׁ֔יו בְּיַרְכְּתֵ֥י הַמְּעָרָ֖ה יֹשְׁבִֽים
And he came to the sheepfolds along the way. There was a cave there, and Saul went in to relieve himself. Now David and his men were sitting in the back of the cave.
[Refer to Gemara Brachos 62b which brings this instance and teaches that there was a fence within a fence, and a cave within a cave, and thus Shaul entered to defecate in the interest of modesty.
In addition, this attribute of modesty is also highlighted in masechta Megilla 13b where it notes that this modesty resulted in him being the ancestor of Esther. The gemara goes on to explain that this modesty was the fact that he did not reveal to his uncle that he had been anointed as king (Shmuel I 10:16).
- He wasted his own money so that the rest of the nation wouldn't have to - Refer to Shmuel I 11:6-7
וַתִּצְלַ֤ח רֽוּחַ־אֱלֹהִים֙ עַל־שָׁא֔וּל בשמעו [כְּשָׁמְע֖וֹ] אֶת־הַדְּבָרִ֣ים הָאֵ֑לֶּה וַיִּ֥חַר אַפּ֖וֹ מְאֹֽד׃ וַיִּקַּח֩ צֶ֨מֶד בָּקָ֜ר וַֽיְנַתְּחֵ֗הוּ וַיְשַׁלַּ֞ח בְּכָל־גְּב֣וּל יִשְׂרָאֵל֮ בְּיַ֣ד הַמַּלְאָכִ֣ים...
When he heard these things, the spirit of God gripped Saul and his anger blazed up. He took a yoke of oxen and cut them into pieces, which he sent by messengers throughout the territory of Israel...
Also refer to the Midrash Shocher Tov 7:2 which notes that he freely spent money on Israel's wars.
- He ate chullin in a state of purity - See Shmuel I 20:26
וְלֹֽא־דִבֶּ֥ר שָׁא֛וּל מְא֖וּמָה בַּיּ֣וֹם הַה֑וּא כִּ֤י אָמַר֙ מִקְרֶ֣ה ה֔וּא בִּלְתִּ֥י טָה֛וֹר ה֖וּא כִּֽי־לֹ֥א טָהֽוֹר
That day, however, Saul said nothing. “It’s accidental,” he thought. “He must be unclean and not yet cleansed.” (Refer to Rashi there)
This point is repeated in the Midrash Shocher Tov 7:2 above.
These last two points are brought in Bamidbar Rabbah 11:3 - שֶׁהָיָה עָנִי וְאוֹכֵל חֻלִּין בְּטָהֳרָה, וּמְבַזְבֵּז אֶת מָמוֹנוֹ וְחָס עַל מָמוֹנָן שֶׁל יִשְׂרָאֵל
- He sanctified all that he dedicated - Divrei HaYamim I 26:28
וְכֹ֨ל הַֽהִקְדִּ֜ישׁ שְׁמוּאֵ֤ל הָרֹאֶה֙ וְשָׁא֣וּל בֶּן־קִ֔ישׁ
All that Samuel the seer had dedicated, and Saul son of Kish,
- He equated his own honour to that of his servants - Shmuel I 20:25
וַיֵּ֣שֶׁב הַ֠מֶּלֶךְ עַל־מ֨וֹשָׁב֜וֹ כְּפַ֣עַם ׀ בְּפַ֗עַם אֶל־מוֹשַׁב֙ הַקִּ֔יר וַיָּ֙קָם֙ יְה֣וֹנָתָ֔ן וַיֵּ֥שֶׁב אַבְנֵ֖ר מִצַּ֣ד שָׁא֑וּל וַיִּפָּקֵ֖ד מְק֥וֹם דָּוִֽד
When the king took his usual place on the seat by the wall, Jonathan rose and Abner sat down at Saul’s side; but David’s place remained vacant.
See Bamidbar Rabbah 11:3 above - וְשֶׁשָּׁקַל כְּבוֹד עַבְדּוֹ לִכְבוֹד עַצְמוֹ
- He was quick to do teshuva - Shmuel I 26:21
וַיֹּאמֶר֩ שָׁא֨וּל חָטָ֜אתִי שׁ֣וּב בְּנִֽי־דָוִ֗ד
And Saul answered, “I am in the wrong. Come back, my son David
Thus, were it not for his informants coming and telling him the whereabouts of Dovid, he would never have chased after him.