If you insist on taming one without an Ichthythornis, a good thing to be aware of is that the Hesperonis will sometimes be unable to eat a fish it kills, swimming back and forth to try and "find" it.

You can pick up a fish it's killed and feed it to it, and it counts for tame progress.

Also be aware that if you drop a fish you've killed, it can eat your fish off the floor and it will not give tame progress.

More Hesperornis Taming & KO Tips