There are two possibilities for organising your summer camp. You could organise your classes around themes or make project work the centre of your lessons. You could of course, do both. I agree with you that if you have short preparation lessons and then games, it is difficult to link everything into an organised package.
You could choose a theme for each day e.g. music, sports, families, work, different cultures. Your choice will depend on the level of your students, the materials you have available and the students’ interests. You can then choose activities that are linked to the theme; make sure you cover listening (this could be songs and video), speaking, vocabulary, grammar and pronunciation. Of course, you can include games in this and you can include a game at the beginning of each day as a warmer. The warmers don’t necessarily need to be connected to your chosen theme. You can include reading and writing if this is appropriate to the objectives of the camp. A good way to proceed is: warmer, listening, speaking about the listening, language work from the listening or other source, more speaking. Generally, get them speaking English as soon as they come into class; listening goes before speaking and reading goes before writing. By taking a theme approach, your lessons will have a common thread which makes them better organised and the students will be using similar language throughout the day which will help them remember it.
As for projects, you can ask students to work on a particular topic to produce something concrete and tangible. Given the amount of time you have, I suggest you pick the topic of the project; this could be producing the front page of a newspaper, preparing an ad for a video game (for a magazine or television), producing a poster for visitors to their town or country. You will need to pre-teach a certain amount of vocabulary and language structures; this will depend on the theme you have chosen. If you decide to do a project, I think it is best to spend some time each day on it, rather than doing all the work in one day. Remember to include some time at the end of the camp for students to publicise or display their projects to classmates (or summer camp staff). They’ll have worked hard on their projects and it’s nice for them to show it off.
For more ideas on projects, you could look at this thread about intensive courses
If you would like more ideas, please feel free to write in again.