What is the mass in $\vec{\nabla}\cdot \vec{B}_g=\text{something}$ called if it could be non-zero? 
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*Maxwell's equation for gravity has 
$$\vec{\nabla}\cdot \vec{B}_g~=~0,$$ 
see Gravitoelectromagnetism in analogy with the electrodynamics. 
What is the mass called that needs to make these equations symmetric in $\vec{\nabla}\cdot \vec{B}_g=\text{something}$?
There are no magnetic monopoles so far but the electric charge is quantized anyway.

*My other question is: What is preventing the quantization of mass or is it considered quantized as built from elementary particles? 
 A: The last part of the question (v4) is a duplicate of this and this Phys.SE questions and links therein. Here we will only address the first part of the question (v4). 
The formalism of gravitoelectromagnetism (GEM) is traditionally$^1$ derived from a weak field limit 
$$\tag{1} g_{\mu\nu}~=~\eta_{\mu\nu}+h_{\mu\nu}, \qquad h_{\mu\nu}~\ll~ 1,$$ 
of General Relativity (GR), where $\eta_{\mu\nu}$ is a fixed background reference metric, usually taken to be the flat space Minkowski metric. Here the first row/column 
$$\tag{2} h_{0\mu}~=~h_{\mu 0}~=~A_{\mu}$$ 
of the $h_{\mu\nu}$ tensor is identified with the GEM gauge $4$-potential $A_{\mu}$, see e.g. Ref. 1. The main point is that the GEM electric field $\vec{E}$, and the GEM magnetic field$\vec{B}$, i.e. the electric-magnetic field-strength $F_{\mu\nu}$, are defined from $A_{\mu}$ by standard formulas for EM in a fixed curved space, which implement the GEM no-magnetic monopole Maxwell equation 
$$\tag{3} \vec{\nabla}\cdot \vec{B}~=~0 $$
manifestly via a Bianchi identity a la this Phys.SE answer. Here we have implicitly assumed that the background metric $\eta_{\mu\nu}$ doesn't have singularities. In other words, a GEM magnetic monopole in a regular spacetime point of the background metric $\eta_{\mu\nu}$ would violate the weak field framework that GEM traditionally is derived from. On the other hand, it seems that GEM magnetic monopoles may be associated with e.g. NUT-type singularities in $\eta_{\mu\nu}$, see Ref. 3.
References:


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*B. Mashhoon, Gravitoelectromagnetism: A Brief Review, arXiv:gr-qc/0311030.

*R. Maartens and B.A. Bassett, Gravito-electromagnetism, arXiv:gr-qc/9704059.

*D. Momeni, M. Nouri-Zonoz, and R. Ramazani-Arani, Morgan-Morgan-NUT disk space via the Ehlers transformation, arXiv:gr-qc/0508036.
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$^1$ Ref. 2 claims to have developed a fully covariant GEM formalism with magnetic monopoles (hat tip: Michael Brown). 
