Where did they come up with the phrase "Bye Bye?" Why is the same word used twice?

it has to be repeated for those guests that didnt get the hint first time around

to be sure, to be sure!

1) "bye (2)
shortened form of good-bye. Reduplication bye-bye is recorded from 1709, though as a sound used to lull a child to sleep it is attested from 1636.

good-bye
1591, from godbwye (1573), itself a contraction of God be with ye, infl. by good day, good evening, etc."
Source and further information:
http://www.etymonline.com/index.php?search=bye&searchmode=none


2) "Reduplication, in linguistics, is a morphological process by which the root or stem of a word, or part of it, is repeated.
Reduplication is used in inflections to convey a grammatical function, such as plurality, intensification, etc., and in lexical derivation to create new words. It is often used when a speaker adopts a tone more "expressive" or figurative than ordinary speech and is also often, but not exclusively, iconic in meaning. Reduplication is found in a wide range of languages and language groups, though its level of linguistic productivity varies.
Reduplication is the standard term for this phenomenon in the linguistics literature. Other terms that are occasionally used include cloning, doubling, duplication, and repetition."

"English uses some kinds of reduplication, mostly for informal expressive vocabulary. There are three types:
- Rhyming reduplication: claptrap, hokey-pokey, honey-bunny, razzle-dazzle, slim jim, super-duper, teenie-weenie, wingding
Although at first glance "Abracadabra" appears to be an English rhyming reduplication it in fact is not; instead, it is derived from the Aramaic formula "Abəra kaDavəra" meaning "I would create as I spoke")

- Exact reduplications (baby-talk-like): bye-bye, choo-choo, night-night, no-no, pee-pee, poo-poo
Couscous is not an English example for reduplication, since it is taken from a French word which has a Maghrebi origin.

- Ablaut reduplications: bric-a-brac, chit-chat, criss-cross, kitty-cat, knick-knack, jibber-jabber, splish-splash, zig-zag
In the ablaut reduplications, the first vowel is almost always a high vowel and the reduplicated ablaut variant of the vowel is a low vowel. There is also a tendency for the first vowel to be front and the second vowel to be back."
Source and further information:
http://en.wikipedia.org/wiki/Reduplication