Why is a rhyming word beginning with "h" put before another word to create a new term?

Solution 1:

Other people have noticed this pattern. It doesn't seem to require an /r/ or an r-colored vowel (as you've already concluded). Shanthi Nadarajan, in "A Crosslinguistic Study of Reduplication," notes two common patterns of partial reduplication with replacement of onset consonants in English words: "h-C" and "C-w".

For "h-C", Nadarajan mentions the following examples "handy dandy", "hoity-toity". For "C-w", two examples are "bow wow" and "teeny weeny." But as others have mentioned in the comments, there are other partially reduplicated words that have other consonant alternations.

Interestingly, there also seem to be examples of reduplicated words in Malay that follow this pattern, but I don't know if that's just a coincidence. Nadarajan mentions that hina dina is Malay for "commoners" and hingar bingar is Malay for "pandemonium". Hina dina seems to be derived from the word dina "poor."

Apparently, Nils Thuns also studied reduplication in English and noticed this pattern. I found a brief description of his findings on the following web page: Slang Phonology, by Urs Dürmüller.

Why it exists

I found the following paper that attempts to explain this pattern: "The Effects of Production and Perception in the English Partial Reduplication", by Seokhan Kang, 2009 (PDF download).

Kang argues that "partial reduplication is motivated by only one principle: maximize the difference of the perceptual cues. More precisely, rhyme reduplication is motivated by the difference in temporal cues of [noise] and [transition]" (154; 2 in the PDF).

I don't fully understand Kang's argument, but he seems to say that the strong aspiration of /h/ generally enhances a perceptual contrast between the onset of the first and second element of the reduplicated phrase (157; 5 in the PDF). He suggests this contrast might be preferred due to something like the Obligatory Contour Principle (164; 12 in the PDF).

Kang also lists some similar examples from other languages.

Solution 2:

Rather than attempting to explain why so many second-order reduplications begin with h (which I can't, and which suməlic's answer addresses quite well), I just want to confirm the existence of the phenomenon itself, using for reference the list of second-order reduplications that appears in Harold Wentworth & Stuart Flexner, Dictionary of American Slang (1960). This dictionary identifies 94 such terms, of which 23 begin with h:

handsy-wandsy, hanky-panky, heebie-jeebies, Hell's bells, herkimer-jerkimer, hipper-dipper, hocus-pocus, hoddy-doddy, hoity-toity, hokey-pokey, holly-golly, honky-tonk(y), hoop-de-doop, hooper-dooper, hootchie cootchie, hoovus-goovus, hotsie-totsie, huff-duff, hugger-mugger, hully-gully, humpty-dumpty, hunkie dunkie, and hurdy-gurdy

and another 4 begin with an aspirated wh:

wham-bam, whing-ding, whoop-de-do(op), and whooper-dooper

Overall, more than a quarter (28.7 percent) of the slang second-order reduplications listed in Wentworth & Flexner begin with an h sound. The phenomenon is clearly not random (or recent).

This reduplicating tendency doesn't carry over with similar force into first-order reduplications, where words beginning with h sounds account for 6 of the 90 words listed (6.7 percent):

haba haba, haha, ho-ho, housie-housie, hubba-hubba, and hush-hush

or third-order reduplications, where such words account for 4 of the 36 words listed (11.1 percent):

Hot-diggity doggety!, Hot ziggity sack!, whimsy-whamsy, and whim-wham

One noticeable characteristic of the second-order reduplications is that they strongly favor four-syllable constructions over two-syllable (Hell's bells, huff-duff), three-syllable (hoop-de-doop), or six-syllable (herkimer-jerkimer) forms. However, this tendency seems reasonably strong in the non-h second-order reduplications (abba-dabba, boogie-woogie, chiller-diller, ducky-wucky, even-Steven, fuddy-duddy, etc.), too, so it may reflect a broader metrical preference.