What part of Bitcoin’s block genesis can be randomly filled?

Depending on the situation, this may be all or none.

Bitcoin could use any genesis block. Genesis blocks are not regular blocks and are not subject to regular block verification rules. This is the 0th block of the Bitcoin chain, so it is unconditionally valid. The field can be anything, as its content is completely unrelated.

Bitcoin uses certain genesis blocks and not others. However, the genesis block is by definition the 0th block of the Bitcojn chain and there are no other blocks. If you change anything in it, it no longer meets that definition. So you can’t change anything.