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What would be an example of a group $G$ with subgroup $H$ such that $G/H$ is abelian but $H$ is not normal?

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You can only make quotient groups with normal subgroups. Otherwise it is not well-defined.

So it is in fact impossible to give an example of this :)

see http://en.wikipedia.org/wiki/Quotient_group for details.

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    $\begingroup$ It's impossible even if OP replaces "is abelian" with "exists". $\endgroup$ – Gerry Myerson Mar 10 '15 at 12:00

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