The minute we met, as an anniversary gift
The date is shared with everyone who married that day. The minute is not.
The problem with the date
An anniversary date is a good marker and a weak gift. It comes back every year, everyone in the family knows it, and it is shared with several hundred thousand other couples who happened to pick the same day.
The minute is a finer instrument. Nobody else in your life is using it, and it points at a moment rather than a day.
Which minute to take
There is usually more than one candidate: the minute you met, the minute of the ceremony, the minute a message arrived. They are not equivalent, and choosing between them is most of the gift.
The most interesting choice is often not the official one. The ceremony is on the certificate already. The minute before it, or the minute a year earlier when something quieter happened, tends to be the one that lands.
What the recipient actually gets
A minute has an address of its own and a daily turn on the clock. It is a place to point at rather than an object to keep, which suits an anniversary: the thing being marked was never an object either.
Because each minute belongs to one person, it also carries a small, real scarcity. The minute you are thinking of is available until somebody else thinks of it.