I always thought they were just pickled onions. I'd keep the ones below refrigerated.

Here is a link that might be useful: Pickled pearl onions

Here's another that's processed.

Here is a link that might be useful: Pickled onions

The drink served with a cocktail onion is properly called a "Gibson" and the onions used are mild, sweet bermuda onions.
There are lots of different ways of making pickled onions, some even fairly sweet, others more briney and some in a sour vinegar brine.
For your special jar for your friend, I would do the onions like you usually do and put some into a smaller jar. If your friend likes the salted type of onion in his gin, I would dump out your sweet mixture and cover the onions in the jar with white wine vinegar.....or dry vermouth.
The best cocktail onions I ever had were packed in vermouth...but I sure don't know how they made tham.
Linda C