Like others have mentioned, the Mobile Menu isn't editable, and the solutions listed elsewhere tend to be somewhat outdated (GetDiscoveryTools() is no longer present in MinervaSkin.php).
So I made a simple Mobile.js script that did the trick for me. I thought I'd share it in case it helps others.
Here's what the custom links look like — there's a dropdown and a regular version that you can customize. Only drawback is they all use the same watchlist.svg icon (which isn't so bad imo, they just look look like "bookmarked" menu links).
Here's the code I added to Mobile.js:
/* Add to MediaWiki:Mobile.js for custom Mobile Menu links
for MW-1.34.2 with MobileFrontend and MinervaNeue
Just replace span text and href to add links */
var timer = setInterval(function() {
if ($('.menu ul:first').length) {
console.log("mobile menu exists");
clearInterval(timer);
$('.menu ul:first').after(
'<ul class="level1"> \
<li> \
<a href="#" \
class="mw-ui-icon mw-ui-icon-before mw-ui-icon-minerva-watchlist"> \
<span>Dropdown Links</span> \
</a> \
</li> \
<ul class ="level2"> \
<li> \
<a href="/wiki/Link_1" \
class="mw-ui-icon mw-ui-icon-before"> \
<span>Link 1</span> \
</a> \
</li> \
<li> \
<a href="/wiki/Link_2" \
class="mw-ui-icon mw-ui-icon-before"> \
<span>Link 2</span> \
</a> \
</li> \
<li> \
<a href="/wiki/Link_3" \
class="mw-ui-icon mw-ui-icon-before"> \
<span>Link 3</span> \
</a> \
</li> \
</ul> \
</ul> \
<ul> \
<li> \
<a href="/wiki/Second_Link" \
class="mw-ui-icon mw-ui-icon-before mw-ui-icon-minerva-watchlist"> \
<span>Second Link</span> \
</a> \
</li> \
</ul>'
);
$(".menu").find(".level2").hide(); // hide level2 until level1 is clicked
$(".level1").click(function(event){
$(this).find(".level2").slideToggle(500);
}); // if level1 is clicked, dropdown level2
}
}, 100); // check every 100ms
/* Todo: Customize link icons, currently they all use watchlist.svg */
Tested on:
MediaWiki 1.34.2
MinervaNeue (d0be74a) 06:41, 1 October 2019
MobileFrontend 2.1.0 (383273b) 11:42, 7 January 2020
P.S. Apparently folks are working on an official solution for this using mw.util.addPortletLink (see phab:T65459, phab:T231925 and phab:T240910), but it still has some styling issues and I wasn't able to figure out how to implement it for the Mobile Menu. If you know how implement AddPortletLink for this, please do share!