Are there any Jewish historical traditions that young Jewish girls often hoped and desired to be the mother of the messiah?
For this question, I am interested in the historical time period of the 1st century AD or earlier.
Since people are not worshipped in Judaism there's little of a practical benefit of "being the mother of" either in one's lifetime or afterward - we don't see any special treatment of any mother of Judaism's greatest figures - Avraham, Yaakov, Moses, King David, R' Yehuda, Rambam, etc.
However, as starting a prominent dynasty of great Rabbis is great merit for a Jewish woman, I think that every Jewish girl strives to achieve that merit and start a wonderful dynasty.
Not aware of anything like that.
But for whatever it's worth, here's something close -- King David's wives all wished that their respective sons would inherit the throne. King Solomon's mother, Batsheva, actually won by asking for something some-sum, while the others all asked for something zero-sum. Sanhedrin 70a:
כל נשים של בית אביך היו נודרות יהא לי בן הגון למלכות ואני נדרתי ואמרתי יהא לי בן זריז וממולא בתורה והגון לנביאות
[She said to Solomon] -- All the women of your father's house would vow [to G-d, hoping that] "may I have a son fit for the monarchy"; but I vowed and said "may I have a son who is industrious, full of Torah, and worthy of prophecy.
I’m not sure what you mean about “a tradition”. I think every Jewish mother desires that her son should be someone great, but that doesn’t mean they necessarily hope their son should be Moshiach. This of course is ignoring the fact that not every Jewish mother could have such a dream - the Moshiach is to be descended from King David and unless you are married to a man who is of David then you have no luck. Hope this answers your question. There is a good joke that talks about the biggest similarity between Judaism and Christianity:
What’s the biggest similarity between Judaism and Christianity? Every Jewish mother believes their son is the Messiah, and every Jewish man believes his mother is a virgin.