I assume OP is asking about the process of becoming more widely used, not the actual coining of new words/expressions/meanings.
The most common expressions I hear for this process are gaining currency, and perhaps more informally, gaining traction.
There's also gaining acceptance, which I think is less commonly used. Personally I think this applies to a later stage in the process, when a neologism is subject to attacks from linguistic reactionaries seeking to preserve the "pure" language they grew up with.
Thanks to @Optimal Cynic for setting out the various stages in the process...
The original coinage is followed by awareness, currency, traction, acceptance, and finally ignorance (people forget or never knew it was once a neologism).