Hi Kris - Subject to any option plan or similar document you may have, you can structure vesting of options fairly creatively. In your description above, do you want a certain number of options to vest upon every day of work, and for the person to log the time worked?
You could also do something like this: "For this phase of the company, I expect you to work 1 day/week without cash compensation. For your work, I will grant you ____ options in the company, to vest monthly over 4 years, subject to a 6-month cliff (to give us time to start working together to make sure this works for us). If and when your engagement with the company changes (i.e., when you will work more than 1 day/week), we can revise the agreement, increase the options, and so forth."
However, if you think that this person's work schedule will change often (i.e., 1 day/week for a month or two, then two days/week for a month, then back to 1 day/week), then simply specifying a set number of options for each day worked (perhaps with a cap, and always upon a mutually agreed upon work schedule) might make more sense.
These are just some initial, informal thoughts - if you'd like to talk in greater depth, feel free to email me.
Eric F. Galen
Attorney + Entrepreneur