What does γ mean before a gene name?

You're correct: the gamma denotes phosphorylation, as explained and used in this paper's abstract

H2AX is a variant of the H2A protein family, which is a component of the histone octomer in nucleosomes. It is phosphorylated by kinases such as ataxia telangiectasia mutated (ATM) and ATM-Rad3-related (ATR) in the PI3K pathway. This newly phosphorylated protein, gamma-H2AX, is the first step in recruiting and localizing DNA repair proteins.

Yes, in this context $gamma$ indicates phosphorylation. See e.g. "The phosphorylated form of H2AX is called $gamma$-H2AX [18, 19]." from here, and references therein.

I am unsure of the exact basis of this, but I believe that it is because the gamma phosphate group of a nucleotide triphosphate is the one that is generally used by kinases to phosphorylate other functional groups. SEE UPDATE for derivation.

Other notations such as the $p$ prefix are more readable to me, but the usage appears to be cultural and somewhat field-specific.

Update

In response to @MattDMO comment, I read further and found that the terminology derives from this paper. They don't directly explain the terminology, but in context it's pretty clear:

At the maximum, approximately 1% of the H2AX becomes $gamma$-phosphorylated per Gy of ionizing radiation.

In mammals, the serine in this motif is residue 139, the site of γ-phosphorylation.

When mammalian cell cultures are exposed to ionizing radiation and the acid-soluble nuclear proteins are analyzed on two-dimensional AUT-AUC gels, novel components that will be referred to as $gamma$ (Fig. 1, A and B) are found in the H2A region of these gels.

It appears that the name probably derives from an observation that was just "whatever in this 2-D gel is $gamma$-P32 labeled when we induce DSBs when $gamma$-P32 ATP is present."

@MattDMO also points out that the irradiation that induces the gamma component in this context is gamma radiation. So it's all a bit of a muddle, it could also be that.