What's the name for this piece attached to the kitchen faucet hose?
This is a proprietary part and versions of it are common on "pull out head" faucets.  I've installed them from Delta and Moen, and from what I remember they are slightly different. I've never seen them sold as a replacement part from DIY retailers.

Your best bet is to figure out what brand the faucet is and call customer support for a replacement part.  Sometimes companies send out small parts for free because they know they break easily and are proprietary, but usually there's a small cost but much better than buying a whole new faucet!