To quote Wikipedia:

The first noted use of "serendipity" in the English language was by Horace Walpole in 1754. In a letter he wrote to his friend Horace Mann, Walpole explained an unexpected discovery he had made about a lost painting of Bianca Cappello by Giorgio Vasari by reference to a Persian fairy tale, The Three Princes of Serendip. The princes, he told his correspondent, were "always making discoveries, by accidents and sagacity, of things which they were not in quest of". The name comes from Serendip, an old name for Sri Lanka (Ceylon), hence Sarandib by Arab traders. It is derived from the Sanskrit Siṃhaladvīpaḥ (Siṃhalaḥ, Sri Lanka + dvīpaḥ, island).

"Always making discoveries" sounds exactly right for an exploration ship. Plus, the word was created by Horace Walpole in a letter to Horace Mann and Horace obviously is very close to Horatio.

Just "Serendipity" is a bit too short to adequately represent Horatio (and also may run into trademarked troubles), so we add "Fortune" to the name in honor of His wealth!

Fortune of Serendipity