In the prayers of Rosh HaShana, we say hayom harat olam היום הרת עולם. Artscroll translates this phrase as "today is the birth(day) of the world", and offers the following comment:
The phrase is in the present tense, for on Rosh Hashanah of each year the Creation is renewed in its entirety...Although the root הרה usually refers to conception, it it sometimes is used to mean birth. This is its meaning here. Alternatively, according to Rabbeinu Tam, Creation took place on two levels: in Tishrei God decided that He would create the world, and in Nissan He did so. Thus Rosh Hashanah is literally the day on which the world was conceived in God's plan.
I'm more inclined to accept Rabbeinu Tam's explanation (and therefore would prefer a translation like, "today the world was conceived"). I discussed the root הרה here, and I found no examples where it meant birth. (For a more in-depth discussion of the background of and imagery in this piyyut, see these Hebrew articles by Sara Friedland Ben Arza and Yael Levine.)
However, the phrase harat olam has an origin with a very different meaning. It is found in Yirmiyahu 20:17, where the prophet, living at the time of the destruction of the Temple, is cursing the day he was born:
Aside from the negative connotation (which is clearly in contrast to the Rosh Hashana prayer), what stands out here is an entirely different meaning of olam. In Yirmiyahu it means "forever", and in the prayer it means "world". (Note that clearly in the verse here, the root הרה cannot mean "birth", although "pregnancy" is more appropriate than "conception".)
אֲשֶׁר לֹא-מוֹתְתַנִי, מֵרָחֶם; וַתְּהִי-לִי אִמִּי קִבְרִי, וְרַחְמָה הֲרַת עוֹלָם
Because He did not kill me in the womb, so that my mother might be my grave, and her womb pregnant forever.
This transformation took place in the passage from Biblical Hebrew to Rabbinic Hebrew. Many authorities say that all usages of olam in the Bible mean "eternity" or "always". Others find a few examples (Tehillim 89:3, Mishlei 10:25, Kohelet 3:11) where "world" would be a better translation, or at least that the more popular understanding of the verse. Klein mentions that olam might derive from the root עלם, "to hide", meaning "the hidden, unknown time".
It seems that in between the two meanings was a third one - "age, era", parallel to the Greek aeon (the root of the English word "eon"), which had similar meanings, as described here:
Following Biblical Aramaic, 'lm (or similar forms) occur in numerous more recent Semitic languages (Nabatean, Jewish Aramaic, Christian Palestinian Aramaic, Samaritan, Syriac, Mandaic, Ethiopic, Palmyrene, Egyptian Arabic, Arabic). Beginning approximately in the 1st Century A.D., several of these languages start using 'lm in a meaning different from that of the OT, namely, as "world" or "aeon".(The same book describes how the post-Biblical book Ben Sira "stands clearly in a transitional situation with regard to the development of the term olam, with traditional meanings continuing, new ones announcing themselves, and many texts clearly hovering between the old and the new and thus eluding unequivocal determination.")
As suggested here (in Doing Jewish Theology: God, Torah & Israel in Modern Judaism by Neil Gillman), I think this sense is reflected in the Hebrew phrases olam hazeh עולם הזה and olam haba עולם הבא. "This age" or "the age to come" might explain the concepts than the more popular "this world" and "the world to come".
So I think the progress went something like this: "always, eternity" to "long period of time, age"1 to "realm, domain" to "the (entire) universe". (The English word "world" had a similar development.)
We can see the tension between the meanings of olam in this Mishna in Berachot (9:5):
This phrase, מן העולם ועד העולם, is originally found in Divrei Hayamim I 16:36, and is quoted in the Pesukei Dizimra prayers. The different meanings of olam are shown in the various translations. The JPS Tanach has "from eternity to eternity" reflecting the Biblical meaning, the Artscroll adopts the mishna's conclusion with "from This World to the World to Come", and the Koren-Sacks tries to split the difference with "from This World to eternity".
כל חותמי ברכות שבמקדש היו אומרים: עד העולם. משקלקלו הצדוקין ואמרו אין עולם אלא אחד התקינו שיהו אומרים מן העולם ועד העולם
At the conclusion of the benedictions said in the Temple they used at first to say simply, “forever.” When the Sadducees perverted their ways and asserted that there was only one world, it was ordained that the response should be "from world to world” [i.e., two worlds].
Similarly, the familiar phrase melech olam מלך עולם, when found in the Tanach (Yirmiyahu 10:10, Tehillim 10:16) means "everlasting King", but when adopted into the blessings, becomes "King of the World".
I opened with a criticism of Artscroll's translation, but in the spirit of the season, I think this post is very important. Dr. Marc Shapiro starts with a similar approach, criticizing Artscroll for their translation of Adon Olam as "Master of the Universe"2, when he thought "eternal Lord" was more fitting. But in the end, he realized that Artscroll was actually correct. The prayer was written - like the Rosh Hashana prayer - in post-biblical times, and so the poets were thinking of "world", not eternity, when they chose the word "olam".
1. The idea that olam did not always mean "eternity", but rather a limited period of time, is found in a number of Medieval Jewish writings. See Rambam, Moreh Nevuchim 2:28, and Albo, Sefer HaIkkarim, 3:16.
2. See this fascinating article by Philologos, where he discusses how the cartoon "Masters of the Universe" maybe was influenced by the Hebrew Ribbono Shel Olam רבונו של עולם, via Fiddler on the Roof.