Press alt+R in mIRC, click the variables tab and paste the following:

%colors 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%currentcolor 2

Click the remote tab and paste this:

on ^*:text:*:#:{
  if ($cnick($nick).color == $color(normal text)) { 
    .cnick $nick $gettok(%colors,$calc(%currentcolor),32)
    if (%currentcolor == $gettok(%colors,0,32)) { 
      set %currentcolor 1
    }
    else {
      inc %currentcolor
    }
  }
}

If you only want nick colors to apply for people in certain channels, replace # with a list of channels separated by a comma. Example:

on ^*:text:*:#chan1,#chan2:{
  ...
}

STEP 3: Press ALT+B in mIRC, click the nick colors tab and check "Enable nick colors"