Eddystone Light by Amelia Smith

My father was the keeper of the Eddystone Light
Slept with a mermaid one fine night.
From this union there came three:
A porpoise, a porgie and the other was me.

Jack studies accountancy, lives with his maiden aunt in an upright Victorian household, and smells like a fish. The fishy smell drives girls away faster than they can be introduced.

When Jack visits his lighthouse-keeper father on the Eddystone rocks, he meets his mother for the first time since he was a baby. She's a mermaid. Jack's mother sends him on a quest to find his two lost brothers, a porpoise and a porgie fish.

Along the way, Jack falls in love, battles carnival hucksters, and is scolded by a porpoise... and that's before he meets the mermaids.