Group a select number of points in a shapefile based on proximity

Method A
1. Navigate to ArcToolbox > Spatial Statistics Tools > Mapping Clusters > Grouping Analysis.


2. Fill out the following parameters in the Grouping Analysis dialog box.

3. Click OK. The features in a feature class are grouped based on attributes that are as similar as possible.

Method B

1. Navigate to the Tools toolbar, and click the Select Features drop-down > Select By Lasso to manually select the desired points.

Note:
To add more points to an existing selection, hold down the SHIFT key when selecting features. To remove feature(s), hold down the SHIFT key and click the previously selected features. 

To remove all selected features, click the Clear Selected Features tool 

 on the Tools toolbar, or click a non-selectable feature belonging to a layer. 

2. To select the desired points, click and drag the tool to specify the size of the selection extent.


3. Right-click the layer in the Table Of Contents > Data > Export Data to save the selected points as a shapefile.


4. Determine the output of the exported data. Click OK.


If a warning on adding the exported data to the map as a layer is displayed, click Yes.


The selected points are added as a shapefile.


5. To remove the unselected points, remove the shapefile from the Table Of Contents by right-clicking the layers > Remove.

Note:
The Select By Polygon tool can also be used to select the desired features, and steps 1 to 5 can be followed to produce the same result.