The repeating of parts of words to make new forms is called reduplication. There are various categories of this: rhyming, exact and ablaut (vowel substitution). Examples, are respectively, okey-dokey, wee-wee and zig-zag. The impetus for the coining of these seems to be nothing more than the enjoyment of wordplay. The words that make up these reduplicated idioms often have little meaning in themselves and only appear as part of a pair. In other cases, one word will allude to some existing meaning and the other half of the pair is added for effect or emphasis.
New coinages have often appeared at times of national confidence, when an outgoing and playful nature is expressed in language. For example, during the 1920s, following the First World War, when many nonsense word pairs were coined - the bee's knees, heebie-jeebies etc. That said, the introduction of such terms begin with Old English and continues today. Willy-nilly is over a thousand years old. Riff-raff dates from the 1400s and helter-skelter, arsy-versy ( a form of vice-versa), and hocus-pocus all date from the 16th century. Coming up to date we have bling-bling, boob-tube and hip-hop. I've not yet recorded a 21st century reduplication. Bling-bling comes very close - one can't be far away.
Here are a few examples, there are many more:
Rhyming: - These are reminiscent of Cockney rhyming slang, although in that the rhyme is usually with some other unspoken word, which doesn't form part of the phrase. For example, trouble and strife - wife. There are a few exceptions to this guideline, for example, chevy chase - face.
Exact: Several of these are baby-talk, showing how they are used to help youngsters learn the language.
Ablaut: - This appears to be suggestive of a toing and froing motion, i.e. indicating a thing that is this and then that. For example, ding-dong, see-saw, tick-tock, zig-zag. Almost all of these use the vowel 'i' in the first part of the reduplication and either 'a' or 'o' in the second part. Why? I don't know.