Here's one way, but you have to make a copy of your feature class to do it:
Open the "Copy Features" tool in the "Data Management" toolbox.
Select your original feature class as the input, and set the output feature class (this will be a new one).
Click the "Environments..." button at the bottom of the tool dialog.
Click the "down arrows" next to the "Z Values" on the environment settings page.
Under "Output has Z Values", select "Enabled". Click OK to close the environment settings page.
Click OK on the Copy Features tool to copy the features.
The output feature class will be Z-Aware. Note that you can also use the environment settings to set a default Z value, resolution, and tolerance on your new feature class if you want.
The same environment settings are also available from ArcMap's "Geoprocessing > Environments" menu, if you want them to apply globally, and not just to this specific tool.
This does not appear to work if you just copy/paste the feature class from the Catalog window; you need to run a tool like Copy Features.