Rockbox

When the iRiver H300 remote is used, the "Reduce Ticking" setting reduces but doesn't eliminate the ticking noise: it remains too clearly audible during soft passages.

The cause of the raspy, cyclic noise is probably the current that flows through the ground wires when the screen is refreshed.
Since ground impedance is not zero, the current bursts are heard as noise through the phones.

Would it be possible introduce a setting such that screen refresh would stop when backlight went off?

In that case, if one touched the remote's buttons the screen would light up and be refreshed; after timeout, the backlight would be off and the screen refreshing would stop.
Even if (because of a hardware design constraint) the screen would have to be blank, the setting would be desirable - as an user option, it would be left to users choosing it or not.