What is this component called
You will need some SRAM. You could use DRAM, but that would require additional controllers and be way overkill as you don't need all that much capacity. SRAM on the other hand is much simpler but with lower density (less memory per chip), but again as you only need 128bytes of RAM, that is tiny.

If you want to read while writing, you need a dual-port variant. These have basically two address buses so you can access two different values at the same time. For simple dual-port memories, one is a read address, the other is a write address - so you can read from one port and write to the other port independently. There is such a thing called a true dual-port memory, which has two completely independent read/write ports, but you probably don't need that, and it will be harder to come by anyway.

Given you are using it as a frame buffer, you'll need it to be parallel (serial SRAM isn't going to cut it). You can either look for 16bit wide memory, or 8bit wide - if it is 8bit wide, you can simply wire up two of them with the same address and control lines and you have a 16bit wide version.

Basically go looking for a 8 or 16bit wide dual-port SRAM memory with sufficient depth for what you need (doesn't matter if it is deeper than you need, you just don't use the extra words). Pick any search engine or distributer website and start looking - shopping questions are off topic, so I'm not going to go looking for you.