You didn't describe if your rectangle has a fixed ratio of the shorter side over the longer side or if it can be rotated. Here are some ideas that might work for you or give you some leads:
1. Run Buffer tool on the points with an "estimated" buffer distance. You can simply estimate by measuring distances between points on your map where more points are located; take about half of the average distance; or you can se Generate Near Table to find distances from points to other points, followed by other tools to derive a distance.
2. Dissolve the buffers, uncheck "Create multipart".
3. Run Spatial Join to find which buffers CONTAIN the points, specify JOIN_ONE_TO_ONE so you get the Join_Count field in the output.
From here, you can select the buffer(s) with high Join_Count and
- either use Feature To Point tool to get the CENTRIOD point. Put the center of your rectangle at that point. Hopefully it will cover max number of points or pretty close to that.
- or if the shape and orientation of your rectangle is flexible, you can use Minimum Bounding Geometry on the selected buffers, using one of the rectangle geometry types (by width or by area) to see if the resulting rectangles are suitable for your purposes.
