Using @octopus' code with a couple tweaks:
#mobile-navigation-title {
background: url("XXX") no-repeat center;
text-indent: 100%;
height:100px;
}
#mobile-navigation #mobile-navigation-label {
background:transparent;
color:black;
}
You can adjust the mobile navigation button, it was showing up as a black square with grey lines but this way can be a transparent square with black lines. Add it to your Custom CSS Editor. You can upload files directly to the Custom CSS editor for use in the background URL, just replace XXX with the URL of the image file.
I also found that the height didn't expand with the image, so the height needs to be set manually. Use Device View to check how it looks.