This sounds like it could be one of many things. One thought though is to go into the settings of your interactive selection settings menu in the MXD (Selection > Interactive Selection Method) and within that sub-menu make sure the correct option is selected, you'd most likely want Create New Selection. If that got changed it could be actually working correctly by not selecting anything, if for example it was set to remove from current selection, then every time you try and select features it would be trying to unselect those features.
Also, again, I don't know if this is your particular problem or not, but you may want to go to Selection > Selection Options and make sure the the settings there make sense for your use case. One thing of particular note I think for you would be, if your points are actually multi-point features, then you definitely need to check that it is set to select features partially or completely within the box or graphic. If you don't have this set correctly and it's multi-point features, you may be having issues because not all of the points for each feature are within the selection area.
I know these aren't the only two possible causes by any means, and from the steps you say you took checking with other datasets I doubt these are it, but, as I don't know the specifics of your situation, it doesn't hurt to try anyway.
Definitely listen to the two comments on your question though, do run the check geometry tool and, if necessary, the repair geometry. Also, do try viewing it in another MXD; and I do not mean dragging/copying the layer from your current MXD and dragging/pasting it into a new MXD, I mean actually opening a new MXD and going to add data and adding the data from source.
Also, if none of this works, just for testing purposes at least, try converting/exporting the dataset in question to a different format and test it there (ex: if it is currently a shapefile, export to a FileGDB feature class, if it is a FileGDB feature class, export to a shapefile. This way you can at least tell if it is a problem with the data it self, or if it is a problem with just that file.