Yes you can implement your own dropdown showing the list of recent projects. Check these for guide/samples
The problem is that you'll then have two drop-down menus showing a list of projects and the only way you can hide the original menu provided by JIRA might be a bit hacky - you need to create a css file which is included globally on all pages which sets the "display" of the original menu to "none".
For JSM June Challenge #2, share how your non-technical teams like HR, legal, marketing, finance, and beyond started using Jira Service Management! Tell us: Did they ask to start using it or...
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