Rosenthal confirms the theory put forth in the Wikipedia article that the Hebrew name gaga געגע is a duplication of the imperative גע "touch!" (or perhaps better "hit!"). The game was apparently imported to Jewish camps in English speaking countries by Israeli visitors.
The game Ga-ga is not related to the identically spelled געגע - a verb meaning "to long, to yearn". This verb derives from the word געגועים ga'agu'im - "longing, yearning". Klein writes that this word is "probably derived from געה ( = he cried, wailed, wept.)" The verb געה means "to moo, low, bleat", and Jastrow says the meaning "homesickness, longing" is "as the cow lows after her calf".
We find the term ga'agu'im in the Talmud (Shabbat 66b) where it discusses a son who longs for his father - בן שיש לו געגועין על אביו. Rashi translates the term as ברמור"ט, which Steinsaltz says comes from the Old French bramore, meaning "great love".
Now I have found one more location where the term גע גע ga ga shows up, and it is not clear whether it is a relative of the game or of longing.
In the apocryphal Book of Ben Sira, we find the following verse (13:22) -
עשיר מדבר ועזריו רבים ודברוי צכוערים מופים
דל נמוט גע גע ושא ודבר משכיל ואין לו מקום
Now first of all, it should be mentioned that there are a number of different versions of the Ben Sira text in Hebrew, and I do not know which one is the most accurate. One of the ones I found, I believe older than the one quoted above, does not have the same text. But from what I can tell, this version is today considered the most legitimate.
Also, I don't know much about English translations of Ben-Sira, but this one seems to be decent:
When a rich man is fallen, he hath many helpers: he speaketh things not to be spoken, and yet men justify him: the poor man slipped, and yet they rebuked him too; he spake wisely, and could have no place.The translation for גע גע is "they rebuked him". What's the connection? Ben Yehuda writes that the word ga גע here is an exclamation of disrespect and contempt. He offers the following translations (no languages given, but the idea seems clear): "pfui, fi, fie". He says that this is similar to the Talmudic קעקע - the onomatopoeic "to cackle" as found in Kiddushin 31a. He quotes an English translation "hoot", which is both an animal sound, and indicates derision as well. While he doesn't say it explicitly, it would seem to be related to the mooing sound which led to ga'agu'a.
However, Klausner, in his book HaIvrit HaChadasha U'Ba'ayoteha rejects this approach. He says that גע here means "to arrive" (the verb הגיע "arrive" is the hifil of the verb נגע "touch"). The meaning of גע גע ושא is therefore "Go to your place, and take what you have with you, because we don't want you and your things." He says he doesn't understand how Smend came up with the translation "Pfui".
So in summary, looking back I guess I do long for ga-ga and summer camp. But if I'm honest with myself, I think that probably Ben-Sira's advice was true then too. Certainly the rich and popular were more likely to get support when they fell, even in a game of ga-ga...