We see at least 3 names of Sarah in Sefer Breishit.
We see Lucy from what Avraham tells her: לושי ועשי עוגות -- Lucy, and make cakes!
Questions:
Avraham, you've got some 'splaining to do!
While the question is a bit broad and should perhaps be broken up into multiple questions, I will do my best to answer.
The names
As is commonly known, Sarah had 4 names Sarai (Genesis 11:29), Sarah (obviously), Yiskah (Megillah 14a), and Lucy (18:6).
However, she is also clearly identical with Avraham's servant Eliezer. In Genesis (14:14), we read
וַיָּרֶק אֶת-חֲנִיכָיו
He led forth his initiates
As Rashi famously explains, this refers to "Eliezer".
חניכיו - חנכו כתיב זה אליעזר שחנכו למצות
Evidently "Eliezer" must be another name for Sarah, since וַיָּרֶק אֶת-חֲנִיכָיו is numerically equivalent to 505; the numerical value of שרה.
The reasons
Sarah is called Lucy, since Lucy לושי has the numerical value of 346, which is identical to the value of לְאִישָׁהּ; for her husband. Sarah was totally devoted to her husband Avraham, and to his proselytising efforts (cf. Rashi to Genesis 12:5).
She is called Eliezer, in reference to her miraculous salvation (Genesis 12:17) from Pharaoh who kidnapped her (12:16). As the verse in Exodus (18:4) states:
וְשֵׁ֥ם הָאֶחָ֖ד אֱלִיעֶ֑זֶר כִּֽי־אֱלֹהֵ֤י אָבִי֙ בְּעֶזְרִ֔י וַיַּצִּלֵ֖נִי מֵחֶ֥רֶב פַּרְעֹֽה
The name of the primary one was Eliezer for the God of my father was my help, and he saved me from the sword of Pharoh.
The name לושי is actually a compound of לו & שי.
We know that ישמעאל had the first name Lou as seen in Avraham's request 17 18 לו ישמעאל יחיה לפניך.
The word שי means a present or a gift.
Sarah was given this name at this time of her doing chessed by baking bread as a way to spiritually combat the chessed Yishmael did mentioned in passuk 7: ויתן אל הנער וימהר לעשות אתו.
In this way she was able to convince Avraham to reject Yishmael and throw him out of the hous as mentioned in 21 beginning with passuk 9.
To answer your second question, she also was named. חיה. Twice. That is why she is called חיי שרה.
I'd just like to point out that in Bereishis 18 several other names are presented:
וְאַבְרָהָ֤ם וְשָׂרָה֙ זְקֵנִ֔ים בָּאִ֖ים בַּיָּמִ֑ים חָדַל֙ לִהְי֣וֹת לְשָׂרָ֔ה אֹ֖רַח כַּנָּשִֽׁים׃
And Avraham and Sarah were old, coming of age. Sarah had ceased to be Orach KaNashim. (v. 11)
So she used to be called Orach KaNashim but at some point lost that title.
וַתִּצְחַ֥ק שָׂרָ֖ה בְּקִרְבָּ֣הּ לֵאמֹ֑ר אַחֲרֵ֤י בְלֹתִי֙ הָֽיְתָה־לִּ֣י עֶדְנָ֔ה וַֽאדֹנִ֖י זָקֵֽן׃
And Sarah, Bekirbah, laughed, saying: After [I was] Belosi I will be Edna? And my master is Zakein! (v. 12)
So she was also called Bekirbah. It seems that she was also known as Belosi and was expressing astonishment that on top of all of her names she would also be called Edna. To top it off, even Avraham had a bunch of names - he was called Zakein!
HaShem, for the sake of peace (Yevamos 66), decided to not tell Avraham that Sarah had given him such a disparaging name and instead gave it to Sarah, along with another name:
וַיֹּ֥אמֶר יְהוָ֖ה אֶל־אַבְרָהָ֑ם לָ֣מָּה זֶּה֩ צָחֲקָ֨ה שָׂרָ֜ה לֵאמֹ֗ר הַאַ֥ף אֻמְנָ֛ם אֵלֵ֖ד וַאֲנִ֥י זָקַֽנְתִּי׃
And HaShem said to Avraham: What is this that Sarah laughed, saying, even [as] Umnam will I give birth? And I am Zakanti! (v. 13)
Sarah mistook that as Tzachakti, not Zakanti. Scared that people would make fun of her for having a name meaning "I laughed," she tricked the Malachim into changing it for her:
וַתְּכַחֵ֨שׁ שָׂרָ֧ה ׀ לֵאמֹ֛ר לֹ֥א צָחַ֖קְתִּי כִּ֣י ׀ יָרֵ֑אָה וַיֹּ֥אמֶר ׀ לֹ֖א כִּ֥י צָחָֽקְתְּ׃
And Sarah denied it, saying, I am not Tzachakti, for she was scared. And [the angel] said, no, you are Tzachakt. (v. 15)
In summary, this passage reveals seven more names for Sarah, one of which she lost, as well as one more for Avraham.
She was called Lucy basically because she was the first woman. You may think Chava or "the other" (qabalistic personage) was actually the first woman, but actually Sarah Imeinu was the foremother of all humanity which is why we [all] call her "our mother." Hence the earliest female hominine skeleton would obviously have to be her.
Controversial!
You ask, "Why was she called Lucy?"
"Lucy" acquired her name from the song "Lucy in the Sky with Diamonds" by the Beatles, which was played loudly and repeatedly in the expedition camp all evening after the excavation team's first day of work on the recovery site. After public announcement of the discovery, Lucy captured much public interest, becoming a household name at the time.
You ask, "What other names did she have?"
In Ethiopia, the assembly is also known as Dinkinesh, which means "you are marvelous" in the Amharic language.
OK.
You ask how these various names relate to her multiple lives, and I think it is pretty obvious that the fossil and the name Lucy were from Sarah's first life and not from the second, in which all of her stuff in Bereishis happens. This does a good bit to clear up the controversy about Biblical vs. scientific timelines, evolution, etc., and for that you can thank me.
Happy Purim!
יסכה זוֹ שָׂרָה, עַל שֶׁם שֶׁסּוֹכָה בָרוּחַ הַקּוֹדֶשׁ וְשֶׁהַכֹּל > סוֹכִין בְיָפְיָהּ; וְעוֹד יִסְכָּה לְשׁוֹן נְסִיכוּת, כְּמוֹ שָׂרָה > לְשׁוֹן שְׂרָרָה: > > יסכה JISCAH — This was Sarah; she was also named Jiscah (from a root > meaning “to see”, “to look”) because she could see the future by holy > inspiration, and because everybody looked (gazed) at her beauty > (Megillah 14a).
...From this we understand that "Lucy" (-ל+"see") is simply a kinnui of Yiscah. It is also, apparently, a calque of that name. Thus "Lucy" is not a different name from "Yiscah," but merely the same name in a different language.
See here for another possible name for Sarah/Lucy.
The Stone Edition chumash explains:
"Maharal comments on Sarai's two names. A woman has two missions in life, the first from birth as an individual, and the second when she marries and is elevated to a higher, joint mission with her husband. Thus Lucy is the name indicating her personal greatness and Sarai/Sarah, the name indicating her Abrahamitic mission, is used exclusively from the time of her marriage."
Well, no it is not, but you get the gist. The perush is explaining that Sarah did, in fact, have two lifetimes, and possibly more.
