I happened to come across this question again today. In some cases at least, the Hilbert class field $H$ of an abelian extension $K$ of $\mathbf{Q}$ will have to be abelian over $\mathbf{Q}$ for purely algebraic reasons.

Let $F$ be any field, $K|F$ an abelian extension of group $G=\mathrm{Gal}(K|F)$ and containing a primitive $n$-th root of unity for some $n>1$, $\omega:G\to(\mathbf{Z}/n\mathbf{Z})^\times$ the cyclotomic character giving the action of $G$ on $\mu_n$, and $H|K$ an abelian extension of exponent dividing $n$. Then $H=K(\root n\of D)$ for some subgroup $D\subset K^\times/K^{\times n}$, by Kummer theory. It can be checked that $H|F$ is galoisian if and only if $D$ is $G$-stable. When such is the case, the conjugation action of $G$ on $\mathrm{Gal}(H|K)$ coming from the short exact sequence $$ 1\to\mathrm{Gal}(H|K)\to\mathrm{Gal}(H|F)\to G\to1 $$ is trivial if and only if $G$ acts on $D$ via $\omega$. In this situation ($H=K(\root n\of D)$ for some subgroup $D\subset(K^\times/K^{\times n})(\omega)$), a sufficient condition for $H$ to be abelian over $F$, is that the order of $G$ be prime to $n$, because then $\mathrm{Gal}(H|F)=\mathrm{Gal}(H|K)\times\mathrm{Gal}(K|F)$.

I'm sure this situation can be realised when $F=\mathbf{Q}$, for example when the finite abelian extension $K$ has odd degree $[K:\mathbf{Q}]$, $n=2$, the class group of $K$ killed by $2$, and $H$ is the Hilbert class field of $K$. In this case the extension $H|\mathbf{Q}$ will be necessarily abelian.

I happened to come across this question again today. In some cases at least, the Hilbert class field $H$ of an abelian extension $K$ of $\mathbf{Q}$ will have to be abelian over $\mathbf{Q}$ for purely algebraic reasons.

Let $F$ be any field, $K|F$ an abelian extension of group $G=\mathrm{Gal}(K|F)$ and containing a primitive $n$-th root of unity for some $n>1$, $\omega:G\to(\mathbf{Z}/n\mathbf{Z})^\times$ the cyclotomic character giving the action of $G$ on $\mu_n$, and $H|K$ an abelian extension of exponent dividing $n$. Then $H=K(\root n\of D)$ for some subgroup $D\subset K^\times/K^{\times n}$, by Kummer theory. It can be checked that $H|F$ is galoisian if and only if $D$ is $G$-stable. When such is the case, the conjugation action of $G$ on $\mathrm{Gal}(H|K)$ coming from the short exact sequence $$ 1\to\mathrm{Gal}(H|K)\to\mathrm{Gal}(H|F)\to G\to1 $$ is trivial if and only if $G$ acts on $D$ via $\omega$. In this situation ($H=K(\root n\of D)$ for some subgroup $D\subset(K^\times/K^{\times n})(\omega)$), a sufficient condition for $H$ to be abelian over $F$, is that the order of $G$ be prime to $n$, because then $\mathrm{Gal}(H|F)=\mathrm{Gal}(H|K)\times\mathrm{Gal}(K|F)$.

I'm sure this situation can be realised when $F=\mathbf{Q}$, for example when the finite abelian extension $K$ has odd degree $[K:\mathbf{Q}]$, $n=2$, the class group of $K$ has order ($1$ or) $2$, and $H$ is the Hilbert class field of $K$. In this case the extension $H|\mathbf{Q}$ will be necessarily abelian.