In this particular instance, we have a working app running and the basic design is done, but it needs fine tuning and ongoing love. In general I'm wondering what a fair-shake would be for a partner in starting a new project. I'm also interested in the general case, too, of a brand-new startup.
I believe that you can give equity that vests over time, so if you need to let the designer go he/she isn't tied in too much, only gets a small amount of equity.
Their position matters little.
The only thing you can really do is equate the risk/value proposition in a way that seems fair for the amount of risk they are putting in. Are they working without pay? Are they doing this part time? Are they making suggestions that are helping you to decide the direction of the company? All of these are variables that you need to weigh.
Offering a piece of your company is a not normally a form of primary employee compensation. If you are looking at the designer as simply an employee, you'd be better off paying them, otherwise, you should IMO, not consider your designer as simply an employee.
Does that make more sense?