exhaust, use up, run out of, deplete, befriend, serve
盡 centers on reaching the absolute limit of something: using it up completely, depleting resources, or exhausting effort. The less common senses of befriending or serving can be seen as giving everything of oneself to another.
盡 is the traditional form of 尽. It originally depicted a hand holding a brush over a vessel, suggesting cleaning or emptying it completely, which led to the core meaning of exhausting or using up.
Picture a hand holding a brush (聿) over a vessel (皿), scrubbing it until every last drop is gone. That's the idea of exhausting or using something up completely.
For ジン, imagine a gin bottle being emptied to the last drop: gin -> ジン, and the bottle is completely exhausted.