H-1B nonimmigrant status has so many restrictions, you might be surprised to learn how unrestrictive the rules are when it comes to:
Practical limits exist, of course—you probably don't want to take on five half-time jobs in the United States. But as far as the H-1B rules are concerned, you can work full- or part-time for one, two, or several employers, so long as each job qualifies as an H-1B occupation and each employer is willing to take on the H-1B obligations.
This flexibility creates interesting possibilities as well as some procedural puzzles. (For a general account of the H-1B requirements, see H-1B Visa to the U.S.: Who Qualifies?.)
The U.S. Department of Labor requires employers to pay H-1B workers at least the going rate ("prevailing wage") in their area for the type of work they do, and more if the employer pays more to other workers doing the same job. This requirement applies to part-time as well as to full-time jobs. Thus, though you can work any number of hours, and a variable number of hours per week or per month, each employer must pay you at an hourly rate that satisfies this wage rule. The wage rate and hours for part-time jobs can be tricky to represent in the H-1B paperwork; you and your employer might want to consult an immigration attorney for help.
Though you can work any number of hours in H-1B status without offending the H-1B rules, a change from full- to part-time work or any other significant change in the number of hours you're working for a given employer requires filing an amended visa petition with USCIS.
The H-1B rules impose a yearly limit or "cap" on new H-1B petition approvals. Because of this cap, many H-1B eligible workers end up having to wait for an employer to be able to file a petition for them and then, after the petition is approved, having to wait for the new fiscal year to begin (October 1) before they can work.
If you find a second H-1B job when you are already employed in H-1B status, however, your new H-1B employer will be able to petition for you right away, even if the yearly supply of H-1B slots has already been used up. This is because you have already been counted against the cap.
Some employers—primarily government research institutions, nonprofit organizations, and universities—enjoy an exemption from the yearly cap. They can petition any time for a new H-1B worker. If you are working for one of these types of institutions when you find a second H-1B job, however, your new employer can petition immediately, without regard to the yearly cap, even though the new employer is not itself exempt from the cap. However, this works only for so long as you continue your employment with both employers.
Your overall stay in H-1B status is generally limited to six years (though important exceptions apply if your employer has started the green card process for you— see How Long an H-1B Worker Can Stay in the United States). It would be nice if part-time work counted proportionately against this six-year limit, so that, for example, you could stay 12 years if you worked only half-time. But that isn't the way it works: What counts is your actual period of stay in the U.S. in H-1B status, not how much H-1B authorized work you've done.
However, if you have actually been staying outside the U.S. part of the time, coming to the U.S. periodically to work in H-1B status, then you can add the time you were outside the U.S. back onto the end of your authorized stay to stretch it beyond six years. What is more, if your H-1B job is seasonal or intermittent, or if you spend less than six months out of a year in the U.S., then you can maintain your H-1B status in the U.S. indefinitely, without regard to the six-year limit.
Part-time workers often work on contract—but this won't work for H-1B workers. The H-1B rules specifically require an employer-employee relationship. You should therefore insist that you be on the employer's payroll and receive a W-2 to prove it, as you might well need to show this later on in extending your H status or applying for a green card.
An argument could be made that the method of payment doesn't determine whether an employer-employee relationship exists, but frankly this isn't an argument you'd want to have to make to USCIS. The wise course is to insist on W-2s and reject 1099s.
Part-time work often goes with study, and nothing in the rules prohibits an H-1B worker from enrolling in school. In fact, you can work full-time in H-1B status and also attend school, if you are up to it. The practical question is whether you can juggle work and school such that you satisfy the terms of your H-1B employment, because that is what keeps you in status.