How about a Parallella, it has HDMI output and USB host for keyboard and mouse.

This website has a lot of suggestions. The zynq SOC is also used in the Parallella:

https://joelw.id.au/FPGA/CheapFPGADevelopmentBoards

The miniZed board⁽¹⁾ uses a Xilinx chip, but it doesn't seem to come with a license for software except for SDSoC⁽²⁾, which doesn't seem to provide any software for HDL, just C++ and OpenCL.

[1] https://www.xilinx.com/products/boards-and-kits/1-odbhjd.htm...

[2] https://www.xilinx.com/products/design-tools/software-zone/s...

The terasic de0-nano SoC is pretty accessible. It runs Linux and has a decently sized fpga for the price.

Interesting suggestions, thanks!