Origin of “riff”
Solution 1:
They aren't lying when they say it's uncertain.
Another possibility is that of riffing. It's generally considered that the verb riff came from the noun riff, and the gerund riffing from that by the normal -ing production, but the gerund riffing is in fact found slightly earlier. This though just moves the problem around; we now ask where riffing came from, and if it didn't come from riffing < riff(v) < riff(n), then it we've more or less the same possibilities of refrain, ripple, etc.
To consider how likely they are, let's look at a couple of early uses. The noun form first:
1934 Tune Times Aug. 601/1 He runs through the gamut of negro piano riffs.
1962 J. Baldwin Another Country i. i. 16 They might swap stories of..gigs they'd played, riffs they remembered.
The verb:
1935 Atlanta Daily World 5 July 2/1, I..got one of those mouthpieces I invented which has never been used in this country before. It takes an iron lip and an iron jaw to ‘riff’ through 'em.
1948 S. Finkelstein Jazz 213 A single instrument..could riff as effectively as, and even more subtly than, a full band or full choir.
The gerund:
1933 Pittsburgh Courier 15 July ii. 2/6 They pay to hear..riffing of saxophones.
1949 L. Feather Inside Be-bop vi. 42 Jo Stafford's arrangement of The Gentleman is a Dope began with four bars of unmistakably bop riffing.
1958 G. Boatfield in P. Gammond Decca Bk. of Jazz xxiv. 310 These are extraordinary and unique tracks, with Dodds making soaring and at times agonizing music against the sombre riffing of the brassmen.
I've deliberately ignored figurative uses and kept to some of the earlier only.
Now, if we accept that the noun came first, then looking at the first two cases quoted above, it is quite easy to imagine refrain being substitute in those two uses without much damage to the sentence. It's not quit right to consider later uses of riff as exactly synonymous with refrain, but at this point in history, that seems to work okay. So, the theory is that riff started out as pretty much just an abbreviation of refrain and the went its separate ways.
If however the very or the gerund were actually the first forms, then riffle becomes a more likely possibility, though alas that too is a word of unclear origin and perhaps of multiple origins in different senses. Ripple is a possible origin of riffle in this sense, but not the only one.
Solution 2:
The suffix on riffle is iterative.
There are many examples of this type of derivation in English. One is patter from pat. (It’s defined as a rapid succession of light taps.) Word pairs like pat/patter are an established pattern (haha) in English.
In the case of riff,the original, non-iterated verb had been lost. The riff invented/innovated by jazz musicians is thus a back formation. It follows the pattern, which must have helped it lexicalize (in other words, catch on.)
It makes sense if you understand a riff as the “hook”, the essence of the refrain, what is being repeated. I doubt you will be able to pinpoint one source word - the intersection of possible sources adds to its appeal. The musicians who came up with this were riffing on language, so to speak.