I was setting up a System DSN connection to use in SSRS. I would create the ODBC connection using the SQL Server Native Client 11 driver on the SYSTEM tab of ODBC connections. Then in SSRS, the DSN name would not appear. This is because I needed to create the DSN using the 32 bit ODBC ui, which is not the default on my machine. I had to specifically load it using : C:\Windows\SysWOW64\odbcad32.exe
Now it will show in SSRS.
I had found this on SO:
Are you using a 64-bit machine? If so, be aware that on a 64-bit machine there are TWO ODBC Data Source Administrators, one for 32-bit and one for 64-bit drivers. If you create a System DSN in either one, it is not visible in the other.
The 64-bit version is usually in C:\Windows\System32\odbcad32.exe
The 32-bit version is usually in C:\Windows\SysWOW64\odbcad32.exe
I’m not sure which version Reporting Services is looking for but try opening both versions up and re-creating your DSN in the one where it’s missing (assuming there are both 32-bit and 64-bit drivers for your data source).