I don't believe you can directly change the data type from point to multipoint, but you can create a new feature class of type multipoint, then copy/paste the point features from your point to multipoint feature class. You can then transfer the existing point attributes to the new multipoint geometries using Spatial Join.
* I'm not sure if there is a single tool that will combine the above steps. I would have guessed Append, but that does not work between point to multipoint feature classes.