Two of the most common enhancements you will write are commands and tools.
A typical command is a toobar button. Your command class will receive the OnClick event and you can do what you want from there. There is generally no user interaction with the map.
A typical tool is a toolbar button that sets your tool class as the 'active' tool. There is no more than one active tool at a time. As long as your tool is the active tool it will receive mouse events as the user interacts with the map.
ArcObjects 10 .NET SDK Help Commands and Tools
ArcObjects 10 .NET SDK Help Walkthrough Create a custom Tool
So, you probably want to create one or two tools. If the points will always be 'far' apart you could create one tool that either creates a new point at a click location, or edits any existing point within a specified radius of the click location.
In either case the code is similar, you create a class that inherits from the correct base class and implements the required interfaces and the write the code to handle OnClick e.g. convert the windows coord mouse click position to map coordinates, search for point features within a radius of your point feature, open a dialog to display/edit attributes of your point feature.
Instead of searching withing a radius of the click location you could also setup the snapping environment when your tool is activated.
ArcObjects 10 .NET SDK Help Snapping in ArcGIS 10
