In general this is taken to mean the value of Gonshor's logarithm at $\omega$. This was defined in the tenth chapter of his 1986 book An introduction to the theory of surreal numbers where you can find a justification for my answer.

In an informal way, the function $\ln$ is the "simplest" function that is eventually smaller than each power function $x\mapsto x^r$ for $r \in \mathbb{R}^{>0}$ but eventually greater than any constant function. So if $\ln(\omega)$ could be the simplest number that is greater than each real number but smaller than each power $\omega^r$ for $r \in \mathbb{R}^{>0}$, that would be nice.

Indeed $\ln(\omega)=\{ \mathbb{R} \ | \ \omega^{r}: r \in \mathbb{R}^{>0}\}$. In Conway normal form, this is a monomial $\omega^{\omega^{-1}}$. You can also write $\ln(\omega)=\{ \mathbb{N} \ | \ \omega^{2^{-n}}: n \in \mathbb{N}\}$, then the difference is that the elements in brackets are simpler than $\ln(\omega)$ in the sense of the simplicity relation on surreal numbers.

edit: since each $\omega^{2^{-n}}$ has birth day $\omega.(n+1)$, the birth day of $\ln(\omega)$ is $\omega^2$

In general this is taken to mean the value of Gonshor's logarithm at $\omega$. This was defined in the tenth chapter of his 1986 book An introduction to the theory of surreal numbers where you can find a justification for my answer.

In an informal way, the function $\ln$ is the "simplest" function that is eventually smaller than each power function $x\mapsto x^r$ for $r \in \mathbb{R}^{>0}$ but eventually greater than any constant function. So if $\ln(\omega)$ could be the simplest number that is greater than each real number but smaller than each power $\omega^r$ for $r \in \mathbb{R}^{>0}$, that would be nice.

Indeed $\ln(\omega)=\{ \mathbb{R} \ | \ \omega^{r}: r \in \mathbb{R}^{>0}\}$. In Conway normal form, this is a monomial $\omega^{\omega^{-1}}$. You can also write $\ln(\omega)=\{ \mathbb{N} \ | \ \omega^{2^{-n}}: n \in \mathbb{N}\}$, then the difference is that the elements in brackets are simpler than $\ln(\omega)$ in the sense of the simplicity relation on surreal numbers.

re-edit: my past answer for the birth day was wrong. In fact each $\omega^{2^{-n}}$ has birth day $\omega+\omega^2.n$, so the birth day of $\ln(\omega)$ is actually $\omega^3$.