The surname Rasmus is a patronymic name derived from the German personal name Erasmus (which in turn was derived from the Latinized form of the Greek word "erasmos," meaning love), borne by an early Christian saint.  Generally, many of the family evolved the name to Rasmussen in Denmark. The suffix '-sen' is translatable as 'son of.'
Emigration to New Zealand followed in the footsteps of the European explorers, such as Captain Cook (1769-70): first came sealers, whalers, missionaries, and traders. By 1838, the British New Zealand Company had begun buying land from the Maori tribes, and selling it to settlers, and, after the Treaty of Waitangi in 1840, many British families set out on the arduous six month journey from Britain to Aotearoa to start a new life. Early immigrants include:
Rasmus Settlers in New Zealand in the 19th Century
Niels Rasmus, aged 23, who arrived in Wellington, New Zealand aboard the ship "Halcione" in 1870