= Solution
Use <natural units> and write the common light <constituent quark mass> as $m$, and the strange one as $M$. For two <spin one-half> constituents,
$$
\mathbf S_1\cdot\mathbf S_2=\frac12\left[S(S+1)-\frac32\right]
=\begin{cases}-3/4,&S=0,\\+1/4,&S=1.\end{cases}
$$
The relative hyperfine contributions are $-3$ and $+1$. Absorb their normalization and spatial overlap into pair-dependent parameters $h_{\ell\ell},h_{\ell s},h_{ss}$. The <meson hyperfine mass splitting> model gives
$$
\begin{aligned}
m_\pi&=2m-3h_{\ell\ell},&m_\rho&=2m+h_{\ell\ell},\\
m_K&=m+M-3h_{\ell s},&m_{K^*}&=m+M+h_{\ell s},\\
m_\phi&=2M+h_{ss}.
\end{aligned}
$$
A common color-magnetic coefficient gives $h_{ij}\propto1/(m_i m_j)$, but its numerical value is not needed.
In the specified <Eta octet state> used for the <eta meson> the light-flavor probability is $1/3$ and the strange-flavor probability $2/3$. Assuming a flavor-diagonal <mass> operator,
$$
m_\eta=\frac13(2m-3h_{\ell\ell})+\frac23(2M-3h_{ss})
=\frac23m+\frac43M-h_{\ell\ell}-2h_{ss}.
$$
Inserting the five <mass> expressions into the proposed combination makes the coefficient of $h_{\ell s}$ equal to $4-4=0$. Those of $m,M,h_{\ell\ell},h_{ss}$ are $2/3,4/3,-1,-2$. Consequently
$$
\boxed{m_\eta=-\frac13m_{\pi^0}-2m_{\rho^0}+4m_{K^{*0}}+\frac43m_{K^0}-2m_\phi.}
$$
This is an identity within the assumed <mass> operator and octet flavor assignment. It is not an exact relation among measured <masses>: <eta mixing> and additional spin-independent binding effects are outside those assumptions.
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