Why are option expiry dates not standardized?

Many stock tickers have different option expiry dates. Some have weekly options available, others don't. Some have extremely long dated options available, others don't. On a very basic level this is due to trader interest in the underlying equity

You are correct. Which companies offers weekly options and/or LEAPs is due to trader interest in the underlying equity, aka liquidity.

Why not just make it so that every optionable symbol has a standardized set of expiry dates? The costs incurred by inserting a few more rows into the exchange's database should be insignificant, no?

There are about 4,200 equities that offer options. Only about 500 of them offer weekly options. And then there's another 300 or so ETFs that offer options but not weekly options. And then there's the issue of the 500 or so (a guess) stocks that offer LEAPs. The cost to add 4,000 more weekly options and perhaps the same number of LEAPs would not be insignificant.

... but what is the specific mechanism that determines what expiry dates are available for a particular ticker symbol? Why not just make it so that every optionable symbol has a standardized set of expiry dates?

Once you get past that, options are standardized.

There are 3 expiration cycles for stocks:

• January, April, July, October

• February, May, August, November

• March, June, September, December

All optionable stocks offer options in the current month, the following month and at least the next two months in the cycle.

Of the 4,200+ optionable stocks, about 500 offer weekly options for the next 8 weeks.

A large number optionable stocks offer LEAPs (long term options expiring in more than one year) resulting in two subsequent January expirations.

Stocks and ETFs whose options are heavily traded may have even more expiration months. For example, the most heavily traded options are on the SPY which breaks all of the rules. Its expirations include:

• 16 midweek and weekly expirations
• 8 monthly expirations
• 3 end of month quarterly expirations
• 7 LEAPs expiring in various months

The short answer is that the greater the interest that traders have, the more months and the more strike prices that will be available to trade.

Here is the CBOE explanation of the option cycles.

Here is the CBOE directory of weekly options.