screen mirroring does not list my tv name

If you can't use AirPlay or mirror your device's screen - Apple Support

Not all TVs/displays can mirror directly from an iPhone.

What TV do you have?

~Lyssa

I believe screen mirroring always requires an Apple TV.

AirPlay compatible TVs are just now reaching the market with only the Samsungs available at this time.

You will need to purchase an Apple TV to mirror your phone's screen.

Depending on what you wish to put up there, you may be better off purchasing a Google Chromecast or a Roku device.

~Lyssa

What, exactly, are you trying to put up on the TV? Only certain apps will work with a Chromecast.

You're probably best off reaching out to the TV manufacturer or checking your TV's manual.

~Lyssa