While, to you, this is "good chance to prove" yourself, your boss's job is to put together the team most likely to be successful: The gap from being an individual contributor to leading a team of 4-7 is very significant, and he/she is going to need lots of convincing.
First, I think you need to catalog all the instances where you "acted like" a leader - situations where you resolved conflicts between others; examples where you assigned, reviewed, and re-assigned work to others; instances of evaluating different approaches and picking the best option for the circumstances; examples of managing competing priorities; etc.
Once you have that list of examples, you should look it over and see if you can convince yourself that you actually have what it takes to successfully lead such a project. If you can convince yourself, then essentially that list of examples becomes the core of your argument.
If, after careful consideration, you haven't convinced yourself, one approach to take would be to ask your boss to provide you with opportunities so that you can develop the skill required of a team leader, so that you could be strongly considered for leadership next time. In fact, if you do attempt to convince your boss, and he/she disagrees, I think you should take the same approach: ask for opportunities to develop leadership skills.