How do I remove the participants' name from the video box?

So you typically can't remove the name in the recording once it's there, (likely need some video editing software to try it) but you can prevent it from happening in the future. 

In your user settings for Recording (access via the web), under the Advanced Setting for Cloud Recordings, there's an option you can uncheck for "Display participants' names in the recording".   Uncheck this and hit the Save button, note you have to do this before your next meeting for this to take effect.

Also, if you record to your local machine, I don't believe the name shows up in the corner in the person's video.  At least that's how it used to be, not sure if it still is or not.  

If you don't want peoples' names to show in the corner of their video while the meeting is taking place, in the Zoom app, click Settings, then click Video on the left.  If you scroll down a little there's another checkbox for "Always display participant names on their video" that you can uncheck.

I experienced the same issue when upgrading to the new version. Did you manage to solve this problem when recording to local machine? I have been unsuccessful. 

I'm also experiencing this issue. I  tried to do what @kjhartma said here: If you don't want peoples' names to show in the corner of their video while the meeting is taking place, in the Zoom app, click Settings, then click Video on the left.  If you scroll down a little there's another checkbox for "Always display participant names on their video" that you can uncheck.

But even with the box unchecked, it still shows the person's name in the recording. Anyone else have a solution? Thanks!

I have tried unchecking display names for both cloud and local recordings, and it still shows in the recording, even when I am only recording myself solo.  Would very much like to be rid of this.

It is very frustrating, I use zoom for community who has suffered trauma, the name of my students at the bottom is breach of their privacy, so I after recording the vedio , I play and record it in theOBS studio, which allows you to trim the vedio, I just trim the bottom bi fraction until name disappears and save this recording instead, I understand the extra step but it does the job.

Do it in OBS