Specific word denoting the name of a person who has no last name

Some choices are available. 'Mononym' is given by Collins for a specific circumstance:

mononym (ˈmɒnəʊˌnɪm) n
a person who is famous enough to be known only by one name, usually the first name.

[Collins English Dictionary – Complete and Unabridged, 12th Edition 2014. S.v. "mononym." Retrieved March 16 2016 from http://www.thefreedictionary.com/mononym ]

That is, a famous person known by only one name is called a 'mononym' (according to Collins). This is somewhat far afield, but the circumstance is unusual enough that context and the prefix would probably be sufficient to convey the more general idea of a one-named person, famous or not.

Wiktionary provides another option:

Noun
uninym ‎(plural uninyms)
A single name by which a person or thing is known.

Wiktionary also supplies a source for the neologism:

1985, Jeff Millar, "Ups in engrossing Dragon' overshadow the downs", Houston Chronicle, 19 August 1985: Casting the role with the "uninym-ed" Ariane, a model taking her first shot at acting, is not what you'd call damage control.

(op. cit.)

Four other citations exemplifying use of 'uninym' in the popular press are given, from 2003, 2004, 2006, and 2009.

Of the two, 'mononym' and 'uninym', I would prefer the neologism 'uninym' over the neologistic use of 'mononym', which last has a history in the 19th century defined as "Chiefly Med. Obs. A technical name consisting of one word only" (OED Online). As noted, 'mononym' is obsolete in that sense. 'Uninym' appears to have greater contemporary currency.

Other options include 'monomial' and 'mononomial'. Both are used as nouns and adjectives. Although not obsolete, these virtually equivalent terms have very specific uses historically:

A. n.
1. Math. An algebraic expression consisting of one term only. Cf. mononomial n.
....
2. Chiefly Taxon. A taxonomic or technical name consisting of one word only; = mononym n.