Here is the list of alternative chips:
- ATmega328
- ATmega168
- ATmega8
- ATmega88
- ATmega48
You can use any of the A, P, or PA variants of these chips.
Since they are all pin compatible, any of these can be plugged into the DIP style Uno boards.
The ATmega88 and ATmega48 will require the use of a 3rd party boards platform. I recommend the excellent MiniCore:
Even though the official "Arduino AVR Boards" platform does provide support for the ATmega168 and ATmega8 chips, it is based on older boards that used an obsolete bootloader. So I would even recommend using MiniCore for the ATmega168 and ATmega8.
As mentioned, you will need an ISP programmer to burn the bootloader to any of those chips in order to do uploads as usual vial the USB cable of the Uno. You can use a spare Arduino board as an "Arduino as ISP" programmer if you don't have access to a dedicated ISP programmer.
You should be warned that there is no guarantee that the damage was limited to the ATmega328P chip on the board. The other components on the board are more difficult to replace.