The main purpose of this paper is to study the Hopf bifurcation for a class of degenerate singular points of multiplicity 2 n − 1 in dimension 3 via averaging theory. More specifically, we consider the system x ˙ = − H y ( x , y ) + P 2 n ( x , y , z ) + ε P 2 n − 1 ( x , y ) , y ˙ = H x ( x , y ) + Q 2 n ( x , y , z ) + ε Q 2 n − 1 ( x , y ) , z ˙ = R 2 n ( x , y , z ) + ε c z 2 n − 1 , where H = 1 2 n ( x 2 l + y 2 l ) m , n = l m , P 2 n − 1 = x ( p 1 x 2 n − 2 + p 2 x 2 n − 3 y + ⋯ + p 2 n − 1 y 2 n − 2 ) , Q 2 n − 1 = y ( p 1 x 2 n − 2 + p 2 x 2 n − 3 y + ⋯ + p 2 n − 1 y 2 n − 2 ) , and P 2 n , Q 2 n and R 2 n are arbitrary analytic functions starting with terms of degree 2 n. We prove using the averaging theory of first order that, moving the parameter ε from ε = 0 to ε ≠ 0 sufficiently small, from the origin it can bifurcate 2 n − 1 limit cycles, and that using the averaging theory of second order from the origin it can bifurcate 3 n − 1 limit cycles when l = 1 .