Suggestions for an alternative grading system
I would suggest a rubric that includes the desired qualities of each category.  If you're really worried about the scale, don't include the numbers or letters at all, or change to something like Roman Numerals.  I would not include more than four or five categories, unless you can really break down differences. E.g.,

I. Not making progress. The student needs to improve his/her leadership skills, to include participating in mentor/mentee relationships with junior students, or actively seeking out lead roles in the lab.
II. Pre-leadership.  The student is new to the lab, and has not yet developed leadership roles.  The student should start looking for opportunities for leadership, and should begin considering the type of leader he or she wants to be.  The student should thoughtfully observe other leaders and their leadership styles.
III. Making progress.  The student is actively engaged in a leadership role in the lab and/or has made mentor/mentee relationships with junior students.  The student should continue to develop his/her leadership skills in these roles.
IV. Excellent leadership development.  The student has actively developed his/her leadership opportunities, and is capable of independent leadership in the lab.  Other students view him/her as a quality leader and are comfortable going to the student for guidance.
V. Phenomenal leader.  The student is qualified to independently lead large groups of other students, and is able to mentor others in leadership roles.

In this example, I would start everyone off at (II).  Everyone will know they should look for leadership opportunities, and it should be relatively easy to get into the (III) category.   Obviously, this is only an example, and can be tailored to your situation.

I'm also trying to find something that won't be too discouraging for students whose leadership scores are below average (seeing a 0 or an F would not get the right message across).

I do think having some sort of forcing function is necessary; otherwise, what is the reason to improve?  In my example, you could tone down &quot;Not making progress&quot; (or take it out), but if you do want to show students who aren't improving that they need to, you should have at least one negative category.