That is, if the nature of the work done by the part-time employee is nonexempt in nature under the Fair Labor Standards Act (FLSA), you have to be sure you are paying the worker at least the minimum wage per hour worked according to federal law (or state law if the minimum wage in your state is higher than the federal minimum wage) and that you are paying overtime if the employee is working more than 40 hours in any one workweek to accomplish your assignments.
As a general rule, how you pay part-time employees depends on the kind of work the part-time employee is required to do. The first step will be to properly classify the job under the FLSA as either exempt from the FLSA (e.g., executive, professional, etc.) such as a controller; or nonexempt (e.g., non-key-decision-making role without licensing or advanced knowledge/education requirements) such as an accounts-payable clerk. If you know how to do this accurately, great. If not, obtain input from a human resource expert to help you do it correctly.
It is conceivable that you might employ a part-time executive or professional--for example, a controller. If so, that person is exempt under the FLSA. So in this kind of scenario, you might make an agreement to pay him/her a set amount per week with the stated understanding that he/she will accomplish the tasks agreed upon on an ongoing basis to complete the job responsibilities. Especially with the popularity of teleworking (virtual situations/working from home), this type of arrangement is being used frequently.
However, if the nature of the work is nonexempt from the FLSA--for example, an accounts-receivable clerk--you must be sure to have the person report his/her hours each week and pay him/her for the actual hours worked. Sometimes even part-time workers work more than 40 hours during peak periods, or when other employees are ill or on vacation. In those cases, under federal law, nonexempt employees must be paid time and a half for the hours worked in excess of 40 in any one workweek.
If you are contemplating designating the part-time employee as a contractor, I urge you to carefully review Section Two of IRS Publication 15-A (http://www.irs.gov/pub/irs-pdf/p15a.pdf) You'll see exactly what the IRS considers when determining whether a worker is an employee or an independent contractor (i.e., 1099 worker). In addition, the IRS will help an organization determine the appropriate classification for a worker or group of workers free of charge as regards federal law.
To obtain that service, you simply complete and submit to the IRS Form SS-8 (http://www.irs.gov/pub/irs-pdf/fss8.pdf). However, your state may have more or different requirements.