Update
It is worth pointing out that de Broglie died in 1987 - he thus lived through the development of the quantum theory and his views towards the end of his life were likely much closer to the modern ones, than in 1924, when he formulated his hypothesis. In his Nobel lecture given in 1929 (and freely available to read), he acknowledges the wave-particle dualism (although in careful terms) and admits that his work was superceded by Schrödinger's wave mechanics:

This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. It is based on wave propagation equations and strictly defines the evolution in time of the wave associated with a corpuscle.

Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.

(emphasis mine)

The way de Broglie himself understood his hypothesis is different from how it is understood in modern physics (wave-particle duality): de Broglie thought that there was a wave associated with every massive particle, but didn't really claim that they were one and the same thing. French Wikipedia contains several quotes to this extent:

From the article on Louis de Broglie:

« L’idée fondamentale de [ma thèse de 1924] était la suivante : « Le fait que, depuis l’introduction par Einstein des photons dans l’onde lumineuse, l’on savait que la lumière contient des particules qui sont des concentrations d’énergie incorporée dans l’onde, suggère que toute particule, comme l’électron, doit être transportée par une onde dans laquelle elle est incorporée […] Mon idée essentielle était d’étendre à toutes les particules la coexistence des ondes et des corpuscules découverte par Einstein en 1905 dans le cas de la lumière et des photons. » « À toute particule matérielle de masse $m$ et de vitesse $v$ doit être associée une onde réelle »

Translation:
"The basic idea of [my 1924 thesis] was: "The fact that since Einstein introduced photons into the light wave, it was known that light contains particles which are concentrations of energy incorporated in the wave, suggests that any particle, like the electron, must be transported by a wave in which it is incorporated […] My essential idea was to extend to all particles the coexistence of waves and corpuscles discovered by Einstein in 1905 in the case of light and photons. "To every material particle of mass $m$ and speed $v$ must be associated a real wave"

From article Hypothèse de de Broglie:

« On peut donc concevoir que par suite d’une grande loi de la Nature, à chaque morceau d’énergie de masse propre $m$, soit lié un phénomène périodique de fréquence $\nu_0$ telle que l’on ait : $h\nu_0 = mc^2$  . $\nu_0$ étant mesurée, bien entendu, dans le système lié au morceau d’énergie. Cette hypothèse est la base de notre système : elle vaut, comme toutes les hypothèses, ce que valent les conséquences qu’on en peut déduire. »

Translation:
“We can therefore conceive that, as a result of a great law of Nature, to each piece of energy of proper mass $m$, is linked a periodic phenomenon of frequency $\nu_0$ such that we have: $h\nu_0 = mc^2$. $\nu_0$ being measured, of course, in the system linked to the piece of energy. This hypothesis is the basis of our system: it is worth, like all hypotheses, what the consequences that can be deduced from it are worth. »

The way de Broglie himself understood his hypothesis is different from how it is understood in modern physics (wave-particle duality): de Broglie thought that there was a wave associated with every massive particle, but didn't really claim that they were one and the same thing. Thus, in terms of the OP, it is the second option: There is a wave (of something) moving around the particle. (In de Broglie's own terms the wave is carrying the particle=the particle is transported by a wave.)

French Wikipedia contains several quotes to this extent:

From the article on Louis de Broglie:

« L’idée fondamentale de [ma thèse de 1924] était la suivante : « Le fait que, depuis l’introduction par Einstein des photons dans l’onde lumineuse, l’on savait que la lumière contient des particules qui sont des concentrations d’énergie incorporée dans l’onde, suggère que toute particule, comme l’électron, doit être transportée par une onde dans laquelle elle est incorporée […] Mon idée essentielle était d’étendre à toutes les particules la coexistence des ondes et des corpuscules découverte par Einstein en 1905 dans le cas de la lumière et des photons. » « À toute particule matérielle de masse $m$ et de vitesse $v$ doit être associée une onde réelle »

Translation:
"The basic idea of [my 1924 thesis] was: "The fact that since Einstein introduced photons into the light wave, it was known that light contains particles which are concentrations of energy incorporated in the wave, suggests that any particle, like the electron, must be transported by a wave in which it is incorporated […] My essential idea was to extend to all particles the coexistence of waves and corpuscles discovered by Einstein in 1905 in the case of light and photons. "To every material particle of mass $m$ and speed $v$ must be associated a real wave"

(emphasis mine)

From article Hypothèse de de Broglie:

« On peut donc concevoir que par suite d’une grande loi de la Nature, à chaque morceau d’énergie de masse propre $m$, soit lié un phénomène périodique de fréquence $\nu_0$ telle que l’on ait : $h\nu_0 = mc^2$. $\nu_0$ étant mesurée, bien entendu, dans le système lié au morceau d’énergie. Cette hypothèse est la base de notre système : elle vaut, comme toutes les hypothèses, ce que valent les conséquences qu’on en peut déduire. »

Translation:
“We can therefore conceive that, as a result of a great law of Nature, to each piece of energy of proper mass $m$, is linked a periodic phenomenon of frequency $\nu_0$ such that we have: $h\nu_0 = mc^2$. $\nu_0$ being measured, of course, in the system linked to the piece of energy. This hypothesis is the basis of our system: it is worth, like all hypotheses, what the consequences that can be deduced from it are worth. »

(emphasis mine)