Writing a letter of recommendation for your own, new grad student? I have a new PhD student who needs a rather large number of letters of recommendation for a fellowship.  Normally I would not agree to write such a letter without knowing the student for a longer period of time, but at this point she has exhausted her list of other potential letter writers and has no other good option.  Also, I do genuinely believe in this student's potential (otherwise I wouldn't have hired her!), and have had nothing but terrific experiences with her so far.

Question: How do I write a compelling letter, given that I have known this student for only a short amount of time (less than a year)?

Obviously the fellowship is remunerative for me as well, since I wouldn't have to pay her out of my own grants for a few years.  A cynic sitting on the review board might say, "of course the advisor wrote a good letter—they want the money!"  What kind of tangible information might help dissuade a person with this kind of attitude?

(For reference, the fellowship is a big federal graduate fellowship in the US.)
Several months, if you worked with the student, is plenty of time to get an informed impression, and especially so if your impression is good. This is how I would proceed:

Write how you know/met the student. Explain why/how your interaction developed. In your case, how did the student capture your attention, why you hired the student, what were the motives for this.

Then write facts. You say the student is terrific. What are the facts that support that? For instance, "the student solved a difficult problem in short time. The student very quickly understood the material you gave her to read. The student has produced a publication. The student is going to submit some research proposal earlier than expected. The student is extremely effective in running studies". Or, whatever is relevant in your case. At this stage, only list facts which are relevant to support the case of the student. If the student did not yet work with you, but you know the student from teaching her in class, you could go for "top 2 out of 50", "best coursework I have seen" or whatever is relevant there. Don't be tempted to write about things you do not know. If you have few, but promising, facts, limit yourself to state these and make them compelling.

In the next section, most references are expecting a few words outside of the concrete scientific facts, e.g. team work abilities, presentation skills, or something else of relevance that makes clear that the student has potential. Sometimes, hobbies are mentioned, but I typically make that dependent on whether it is relevant to the position/grant. 

Now summarize your impression about the student, supported by the facts listed above. If you are delighted/looking forward to work with the student, do not hesitate to let that be known.

Of course, you would want the money - but, see it from the point of the funding agency: it is, of course, more likely to feel more confident giving the money to a student whose supervisor radiates the message that she is enthusiastic to work with that student. It is perfectly legitimate to support one's own student, and your reference will, of course, be completely open about that.