You can change a bot's name (as well as its description, picture and other details) only through @BotFather.
To change the bot's name:
1. Open the @BotFather bot.
2. Send the /mybots command to @BotFather.
3. On the keyboard below the message, select the username of the bot you need.
4. On the keyboard below the message, select "Edit Bot".
5. On the keyboard below the message, select "Edit Name", enter the new bot name and send the message.
Congratulations! The bot's name has been changed.
Important! If the name has not changed, wait a few minutes or restart your Telegram client.