You can do either a 1099 or a W-2. There is no limitations to the number of W-2s one can have in reporting taxes. Problems occur, with the IRS, when one "forgets" to report income. Even if one holds only one job at a time, people typically have more than one W-2 if they change jobs within the year.
The W-2 is the simplest way to go and you may want to consider doing this if you do not intend to work this side business into significant income.
However, a 1099 gig is preferred by many in some situations. For things like travel expenses, you will probably receive the income from these on a 1099, but you can deduct them from your income using a Schedule C. Along these lines you may be able to deduct a wide variety of other things like travel to and from the client's location, equipment such as computers and office supplies, and maybe a portion of your home internet bill. Also this opens up different retirement contributions schemes such as a simplified employee pension.
This does come with some drawbacks, however. First your life is more complicated as things need to be documented to become actual business expenses. You are much more likely to be audited by the IRS. Your taxes become more complicated and it is probably necessary to employee a CPA to do them. If you do this for primary full time work you will have to buy your own benefits. Most telling you will have to pay both sides of social security taxes on most profits. (Keep in mind that a good account can help you transfer profits to dividends which will allow you to be taxed at 15% and avoid social security taxes.)
So it really comes down to what you see this side gig expanding into and your goals. If you want to make this a real business, then go 1099, if you are just doing this for a fes months and a few thousand dollars, go W-2.