No, it's not dumb at all. I was planning on doing this in our current kitchen remodel, but our floor plan changed so I have room for a full pantry.
My plan was to cover it with a chalkboard and use drawer slides so the chalkboard can slide over.

They did something on DIY Network that was similar, but they also stole some space from the garage to make it deeper. But it is the same concept.

Here is a link that might be useful: DIY Wasted Spaces

They have an indoor herb garden here incorporated into the kitchen, kind of hard to tell but it looks cool:

Link: http://www.designmind.co.za/profiles/blogs/grand-designs-live-2

You know, the very first time I saw those pot, um, herb-growing light boxes in a store, I had the same thought.

I think the key might be obscuring all the mechanics of it. Some of those grow lights are big and clunky. I think if you approach it so that if the herb thingie doesn't work out, you can remove it and the cab will look perfectly normal as a simple windowed display cabinet.

Also think about waterproofing the bottom somehow. Maybe even just a scrap of laminate.

On a small scale, get cute round hanging pots. Asker.
On a bigger scale get any linear container.
With soil it's an indoor "flower box" .
Without soil, it's hydroponic as long as a pump circulates water twice daily.

i looked into this recently because of an Aerogarden style appliance someone invented.
http://ths.gardenweb.com/forums/load/appl/msg0512441922157.html indoor herb gardening. FRIDGE sized hydroponic system.

Here is a link that might be useful: hang pots (hooks on rail) Asker, Ikea.

Not *dumb*, but kinda much ado about very little. Would you ever need the space for a wider fridge/freezer? Would a buyer...someday? An 8" deep space doesn't offer a lot of options. Shelves? Display for children's artwork? Charge phones? I think the plant idea would be cute, but messy.

First, sayde, can we assume that you wouldn't be trawling for uses for your "found" space if you did not already have a broom closet handy?

Second, as an internationally renowned klutz, I have to ask:
If you have a full-depth 8"-wide growing space, is there a practical way to water and harvest?

Third, your kitchen has an unusually lovely quality of natural light. Is the fridge located somewhere that would keep the gro-light from interfering?

Cheers.