The generic answer to this question is “(Re)install ‘Oracle Developer Tools for Visual Studio’ (ODT)”. Simply getting ODP.NET from Nuget or from other sources is not enough to take advantage of Visual Studio integration. You must install ODT also.
Download “ODTwithODAC”:
http://www.oracle.com/technetwork/topics/dotnet/utilsoft-086879.html
Update 4/2018: If you are using Visual Studio 2017 Community edition, please upgrade to version 12.2.0.1.1 or later. That is the first version that works with VS 2017 CE: http://www.oracle.com/technetwork/topics/dotnet/downloads/odacmsidownloadvs2017-3806459.html
As you noticed, sometimes installations get munged. So if you do not see the driver in the dialogs or are getting generic (non-oracle ORA-) errors when you try to use features then you should reinstall ODT.