Add white noise to an isolated electrical node.
Take the voltage from that node and add it to your desired variable/statement/whatsoever.

Another approach would be this:

V(p,n) <+ white_noise(4*`P_K*$temperature*ron, "thermal");  // (A)
if ( V(ps,ns) > thresh )
     V(p,n) <+ ron * I(p,n);  // (A)
else
     I(p,n) <+ 0;

Verilog-A's value-retention rules say that, if you contribute to the flow (I), then all potential contributions are discarded.  Now your white_noise call is not inside the if() statement.

BTW: I personally believe the error from your simulator is incorrect.  white_noise is *not* an analog operator along the lines of idt() or absdelay(); it appears in "4.5 Analysis dependent funtions" rather than "4.4 Analog operators" in the Verilog-AMS language reference manual (2.2).