The Yiddish word for grandmother is "bubbe". (There are many alternate spellings - in my family, for example, we spell it "bobe" and my wife's family spells it "bubby". So both for shalom bayit, and because it seems to be the most popular spelling, I'll stick with "bubbe").
Where does the word come from? I found this William Safire column from 1990 (inspired by a question about the word bubba, which is unrelated.) He discusses the word bubbe, and offers two possible origins:
Buba is a Hebrew word for "little doll" and may have been the source of an affectionate term for a small grandmother; however, the similar baba is also used for "grandma" in Russian and other Slavic languages, which makes the origin uncertain.
Let's look at the first suggestion - that it derives from buba בובה - "doll". Mr. Safire was mistaken on this one (and I imagine heard about it from his readers). The word buba was coined by Ben Yehuda, together with a co-founder of the Vaad Halashon, Haim Kalmi, in a children's books published in 1904. This certainly postdates the Yiddish bubbe. Klein writes in his entry:
Coined by Eliezer ben Yehudah as a Hebraization of Arabic bu'bu, bubbu' (=little child, doll). Compare German Puppe, French poupee.
Klein also connects the same Arabic root with the word bavuah בבואה - "reflection, image". He gives this etymology:
Related to JAram בוביא and perhaps also to Arabic babbat (=little child), understood as a reflection of a man.This book provides another related Hebrew word: bava בבה - "pupil of the eye", used almost exclusively in the phrase בבת עין - bavat ayin, found in Zecharya 2:12 (babat also appears on its own in later Hebrew literature, like in the song Dror Yikra). It literally means "pupil of the eye", but like the English phrase "apple of the eye", has the sense of "something very dear and important". The authors say it is also related to the Arabic root we've seen earlier, and both derive from an Akkadian word "babu". meaning "child, baby". They point out it has a similar sense to the word for pupil in many languages, including English, which is related to the homonym meaning "student" and has the following etymology:
"center of the eye," early 15c. (in English in Latin form from late 14c.), from Old French pupille (14c.), from Latin pupilla, originally "little girl-doll," diminutive of pupa "girl; doll", so called from the tiny image one sees of himself reflected in the eye of another.A Hebrew synonym that the authors don't mention is ishon אישון, also meaning pupil, and similarly is a diminutive of ish איש, and so literally means "little man".
Ben Yehuda quotes scholars that agree with this etymology of bava, He also quotes those who suggest that the word might be related to the Aramaic word for gate or entrance- baba בבא (which we see in the name of the Talmudic tractates Bava Kama, Bava Metzia and Bava Kama - the three "gates" of the order Nezikin) and the Arabic word for gate - bab. In the entry for the root נבב - "to make hollow", Klein writes that baba derives from the Akkadian babu (=door, gate) which is shortened from nebaba (literally: 'hole,aperture'), which is also from the base נבב.
We've jumped from buba and bavuah to bavat and baba. Let's get back to bubbe.
Safire's second suggestion - a Slavic origin - makes much more sense. We see it in the Russian word for grandmother, babushka, which is a hypocorism (a suffix added for endearment) of the word baba, meaning "old woman".
Ultimately, however, there might be a connection between the Semitic roots we saw earlier meaning a child, and the Slavic ones meaning an old woman. As the Online Etymology Dictionary writes in the entry for the English word "babe", the origin of the word is from baby talk, and so in some languages that comes out as a word for children, and in some as a word for older people (adults).
You might be familiar with the phrase bobe mayse - or the Hebrew derivative sipurei savta סיפורי סבתא- meaning a fanciful story, an old wives tale. What might surprise you, though, is that the phrase actually doesn't come from the word bubbe, but rather from the story of Bevis of Hampton. As Philologos writes:
As improbable as it may seem, the bobbe of bobbe mayseh comes from the name of the hero of the 15th-century medieval Italian romance “Buovo d’Antona,” a Tuscan adaptation of the Norman “Beuve de Haumpton” — known in its 14th-century English version as “Sir Bevis of Hampton.”
Read the Philologos article for more detail about the story (as well as this one by Ari Zivotofsky), but to sum up, the 15th century Hebrew linguist and grammarian Eliahu Bachur translated the stories from Italian in to Yiddish as Bovo-Bukh. (He chose the name Bova apparently as a play on the word bava we saw above, ending the poem with the words “This is the end of the tractate of Bova of Antona.”) The adventures of Bova were so fanciful that even after the stories themselves were forgotten, the name was forever saved in the phrase bobe mayse.
Don't believe me? Ask your bubbe!