The Meaning of Shabbat
Shabbat – שבת (sha-bat) “This is the Hebrew word שבת shabbat literally meaning a ‘ceasing’ or ‘stopping.’ This word is always used in the Hebrew Bible for the seventh day, the Shabbat, the day work ceases for the purpose of rest” – Jeff A. Benner
From the following passage, we learn that Shabbat is a time of ceasing work but also a time for “holy” gatherings.
Leviticus 23:3 “Six days work is done, but the seventh day is a Sabbath of rest, a set-apart gathering. You do no work, it is a Sabbath to יהוה in all your dwellings”.
The word “Shabbat” also has a deeper connection to our English word “Seven” than what one might find immediately recognizable. Isaac E. Mozeson, author of The Word, had this to say regarding SEVEN (SHEVA), SHAVOOA (week), and SHABBAT (see scan of “Seven” entry below for details):
Taken from THE WORD: The Dictionary That Reveals The Hebrew Source of English by Isaac E. Mozeson, pg 151.
The Meaning of Shalom
Shalom – שלום (sha-lome) “When we hear the word peace we usually associate this to mean an absence of war or strife but, the Hebrew meaning of the word shalom has a very different meaning. The verb form of the root word is shalam and is usually used in the context of making restitution. When a person has caused another to become deficient in some way, such as a loss of livestock, it is the responsibility of the person who created the deficiency to restore what has been taken, lost or stolen. The verb shalam literally means to make whole or complete. The noun shalom has the more literal meaning of being in a state of wholeness or with no deficiency. The common phrase shalu shalom yerushalayim (pray for the peace of Jerusalem) is not speaking about an abcense of war (though that is part of it) but that Jerusalem (and by extension all of Israel) is complete and whole and goes far beyond the idea of “peace”.” – Jeff A. Benner
“Shabbat Shalom” together
So when “Shabbat Shalom” is put together, it is generally intended to mean “may your day of no work be peaceful”, but it could have similar variants such as “may you become whole during your ceasing of laborious work” or “may your wholeness be restored as you cease work on the seventh day”.