There are five ways a nonprofit in D.C. can be organized: as a corporation (most popular by far), an unincorporated association, a trust, an association, or an LLC. Each one has specific laws you must know and follow about how to name your organization. Be sure to look those up, and know which ones apply to you, before you get too far down the line.
Like many other states, Washington D.C. also requires that each business have a unique and distinct name. But unlike most other states, it allows you to file for a name reservation before you’re ready to officially incorporate. It’s a good option if you’ve already done the research into a name but don’t have all your documents together yet. These reservations last 120 days and can be renewed online.