What is considered the standard pay for independent caregivers who work on a holiday such as Memorial Day?

The agency I use does time & a half.

Probably depends on your area. Double time on Long Island is not at all surprising. In my area holidays are time and a half.

I was given a rate of double time for the holidays of Christmas Day, New Years Eve Day and New Years Day, this will be very steep. Is this unusual? We are in LI, NY

I can understand a wage increase for working Thanksgiving and Christmas but Memorial Day is a different matter, as would be Labor Day, Presidents Day, MLK Day.... Thanksgiving and Christmas because these are holidays that many families traditionally gather, their only time through the year to get together. Many employers that must be open these holidays will offer comp time off, rather than an increase in wage for working the holiday. Most of the time when wages are increased for holidays it is a incentive for employees to to volunteer to work.

Time and a half. For example, if you pay $20/hour, on a holiday your caregiver would receive the $20 plus half of that = $30/hour. The holiday is midnight to midnight.

It's whatever is stated in your contract. If you have no written contract, then it's just the hourly wage you agreed upon.