  We obtain analytical expressions, both in terms of parametric integrals and
Passarino-Veltman scalar functions, for the one-loop contributions to the
anomalous weak magnetic dipole moment (AWMDM) of a charged lepton in the
framework of the simplest little Higgs model (SLHM). Our results are general
and can be useful to compute the weak properties of a charged lepton in other
extensions of the standard model (SM). As a by-product we obtain generic
contributions to the anomalous magnetic dipole moment (AMDM), which agree with
previous results. We then study numerically the potential contributions from
this model to the $\tau$ lepton AMDM and AWMDM for values of the parameter
space consistent with current experimental data. It is found that they depend
mainly on the energy scale $f$ at which the global symmetry is broken and the
$t_\beta$ parameter, whereas there is little sensitivity to a mild change in
the values of other parameters of the model. While the $\tau$ AMDM is of the
order of $10^{-9}$, the real (imaginary) part of its AWMDM is of the order of
$10^{-9}$ ($10^{-10}$). These values seem to be out of the reach of the
expected experimental sensitivity of future experiments.
