If miners are creating their own mining algorithms or using something standard
The double-SHA256 is the mining algorithm. Miners merely decide which transactions they wish to include in a block, and they an adjust the random nonce field in the block header. Additionally, they create the coinbase transaction (the first transaction in the block), which pays them out for the block reward. There is some additional information they can include in the coinbase transaction which makes no difference to the result, but merely allows them to attach some metadata (such as the name of their mining pool, to indicate they mined it).

The choice of transactions to include can be arbitrary, but the obvious approach is to include them prioritized by highest paying fee, because this maximizes the revenue the miner takes if they are successful.

Everything else is simply random numbers in order to try and find a hash which is below the target. The miners can choose how they generate those random numbers, or how they might partition the range of numbers to attempt between members of a mining pool to prevent duplication of work among the members of the pool.