The entire team can contribute to the 'ideas' part of the project, however what makes a good designer are:
- Excellent communication skills, with the ability to receive and relay information. Designers need to be able to talk with both programmers and artists, and be able to both understand what is being said, as well as have the ability to relay that information in a way that makes sense.
- Extensive knowledge of game systems with the ability to identify the advantages and disadvantages of each.
- The ability to refine game concepts based on this knowledge.
- The ability to take constructive criticism, and to apply it to their work.
- Excellent documentation capabilities.
- And finally, a designer needs to be more than just a designer. You will be much more valued as a designer if you are also a proficient programmer and/or artists. Designers that are simply just 'designers' don't tend to get very far unless they are extremely good at what they do.
I hope that helps!