Stereo "reboots" in 93 Accord

I have a 1993 Accord DX (160,000 miles), which I have fitted with an
aftermarket stereo (Alpine). The problem is, every time the electrical
load changes, the radio "reboots" (cuts out briefly, then restarts).
Once, the entire car went dead when I stepped on the brake, on a stormy
night, in mid-intersection--yikes!

Occasionally, the stereo will reset itself completely, as if the
battery cable had been disconnected. This means reprogramming station
presets and names, EQ settings, preferences, etc., etc. Grrrr...

Most of the time, the stereo seems to act as a "canary in the coal
mine" for voltage glitches, since the rest of the car appears
unaffected; I do notice the "D" indicator and clock will flicker quite
a bit.

Example loads:
--brakes (most common)
--headlights
--engine cooling fan

Most apparent when:
--Just starting out
--Mild weather
--Stop-and-go traffic
--Exiting Interstate after extended period
--A/C is off (oddly, running the A/C minimizes the problem--SOMETIMES)

My guesses:
(1) Bad or missing ground, or corroded battery cable
(2) Weak battery, or intermittent internal short in battery (it's a
cheapo Wal-Mart unit, out of warranty)
(3) Bad voltage regulator and/or alternator
(4) Bad ICU (this was a problem in Preludes and Integras of this
vintage, IIRC, and the Accord uses more or less the same unit)
(5) Bad ELD (expensive!)

 

id check the ignition switch as well. with it running and in park,
wiggle the key around and see if anything bad happens

 

Another suggestion: check that the battery is properly secured and that
no loose metal is flopping around near the positive post. I had similar
symptoms in my '87 Accord, and it turned out the securing clip for the
hood-prop rod was broken, and when I hit the accelerated hard or
cornered JUST so, the rod would make intermittant contact with the
battery post, usually just quickly enough to cause the stereo to drop
out and then come right back on without having a NOTICEABLE effect on
any of the car's systems.

Might check that it's also the proper size battery and that the post
isn't shorting against the inside of the hood.