The first board isn't really a better chance- you have to consider the process as a whole. I will tell applicants to get their scholarship package in early, but that's because the boards work on basically a rolling admissions style. On the first board that you actually make, your application will be evaluated, scored, and briefed, then compared to all other applications at that board (plus all the ones evaluated on previous boards that weren't selected). The top x number will be selected, based on the goal for that particular board. By getting your application in early, your package will have the opportunity to see many more boards than if you wait until, say, January to submit. Applications not selected on the first board will be rolled to the second board, and so on; non selects will not start to be announced until February. So, bottom line, applying earlier just ensures you will have more possible chances for selection, as you will see more boards. Also, if you get the scholarship early, it can have a significant impact on admissions decisions at a number of schools.
Here's the important piece tho- don't rush application submission simply to make an early board. Quality is important, and rushed/poorly written applications are unlikely candidates for scholarships. Ideally, a student should put together a quality application over the summer, and then submit by early September.
Also keep in mind that aplications are only scored once, barring any significant updates. That's why quality is important. Once scored, the score will only change for things like updated ACT/SAT scores, etc. However, don't delay submission because you're waiting for test results or another test session- submitting new results and having them added to your package (and having it re-scored) is simple.
Your application, if not selected at its first board, essentially sits on a big list of all previously boarded applications, with its score next to it. It will have the opportunity to be selected at follow on boards (there are about a dozen throughout the year) if its score is among the top x applications being considered (x is more or less the total number of offers planned, divided by the total number of boards, accounting for students who decline their offers over the course of the process).
It's a confusing process. Hopefully that somewhat answered your question and your head didn't explode.