What is the source for the greeting "שבת שלום"?
Where is it earliest attested in print?
What does it mean?
It's at least as old as the Sh'la (around 1600), according to Taame Haminhagim, kuntres acharon 94 to paragraph 396, which says in the Sh'la's name, via the Baer Hetev OC 307:2: "Someone visiting his friend on Shabas should not say, e.g., 'good morning' as on a weekday, but rather 'shabas shalom' or 'shabas tov', to fulfill 'zachor es yom hashabas'."
It may, of course, be older; perhaps someone else has more info.
וְהִתְקִינוּ, שֶׁיְּהֵא אָדָם שׁוֹאֵל אֶת שְׁלוֹם חֲבֵרוֹ בַּשֵּׁם, שֶׁנֶּאֱמַר (רות ב) וְהִנֵּה בֹעַז בָּא מִבֵּית לֶחֶם, וַיֹּאמֶר לַקּוֹצְרִים יְיָ עִמָּכֶם, וַיֹּאמְרוּ לוֹ, יְבָרֶכְךָ יְיָ
And they corrected this, that one shall inquire after the peace of his friend with the Name [of God], as it says, “And behold, Boaz came from Bethlehem and said to the harvesters, ‘God be with you’, and they said to him, ‘God bless you.’” (Ruth 2:4) (Sefaria translation)
Or in Avot 4:15:
רַבִּי מַתְיָא בֶן חָרָשׁ אוֹמֵר, הֱוֵי מַקְדִּים בִּשְׁלוֹם כָּל אָדָם.
Rabbi Mattia ben Charash says: Be the first to greet every person.
It is the logical conclusion that is this the source of Shabbat Shalom, based on a statement in Yerushalmi (Shabbat 15:3 (78b), quoted in Tosafot Shababat 113b, Aruch Hashulchan 307):
אמר רבי חנינא מדוחק התירו לשאול שלום בשבת
Rebbi Chanina said: Asking shalom on Shabbat was permitted but with difficulty.
The statement refers to how one should change the manner in which one speaks on Shabbat, and to only speak sparingly and holily.