Yeah I agree, probably soundfx got at this first because it's an easy and short bit of code to add a pitch value per frame. Add half your pitch maximum every other frame and you have an octave toggle going. (though not musical, good for drums though)
I'm assuming OP means instrument-based arpeggio like we use today, because note-based has been around forever. (and electronically probably have to dial back to whenever modular sequencers became available for examples in the '40s-'50s) The first instrument-based arps were probably octave based as with sfx, because from a code POV you just need to toggle it on and off every other frame/half your tempo.
If we're talking home electronics I've heard of a few examples bandied about as fairly early. (though all c64 so not extensive) Martin Galway had it in Kong Strikes Back and Bob Landwehr's P.A.S.S music system also does it in a few of the demos. (both 1984) Fred Gray also used octave-based arps extensively and his stuff is quite early too. Looking at a few early sound drivers I wouldn't think it's done with the traditional table-based setup we all use nowadays though. (where pitch/wave tables became general purpose for arps, drums etc.) Probably bespoke code in the driver.
Arcade games tend to add more soundchips to get around the problem (some early '80s games have 2 or 3 AY/SN chips in there) but Do Run Run does have octave-based arps in, and sounds quite different to arcade music from the time. However it's 1984 again so seems a bit late for first in that regard.
This sounds like something goto80 would know. Anders?