Just one thought about usefullness with your design. Give the user the option to mark/remember names. If the user finds a good name at the first page, there is a good chance that the user can not remeber the name from page 1 on page 10.
Good idea. If you click on the name you can tap on "Save" to add it to your saved names list. Does that provide what you wanted?
Make sense now, but maybe you should reduce the clicks(on the desktop) for the "1. open modal -> 2. save/buy click -> 3. close modal" action. There is enough space to the left and right side of the names.
You're right. It is too many clicks to save a name you like. The feature would be more discoverable that way, too.