Here's a quick tip for adding text before the menu button (the one with three lines) in the Divi Theme mobile header. Just add in the following CSS:

.mobile_nav::before { content:'Menu'; vertical-align:top; line-height:2.2em; }

Replace "Menu" with your choice of text and you're good to go. Note that the text will be clickable and will bring up the mobile menu.

Note: adjusting the vertical position of the text is a bit tricky as margins and padding don't seem to have much effect. Setting vertical align to top and adjusting the line-height seems to give the best control.