I'm wondering if it makes sense to plant a cover crop in my raised beds over winter. If so, what plant would you recommend for south-central Ohio.
I have three 4'x8' double dug beds.
I've done it. Basically, the hardest part is turning it under in such limited space, I used a spade. In a raised bed, there are easier ways to improve the soil, like turning in finished compost. Also, the allelopathic effect of most over-winter cover crops means that you'll have to wait 2-3 weeks after turning under for unaffected germination.
You are in south-central Ohio, which has a hardiness zone from 6a-6b. I'm in southeastern Pennsylvania, with an almost identical climate, and a hardiness zone of 6b. I've found that winter rye is the best grower early in spring. I sowed in October, and the seed germinated quickly, growing to the three leaf stage before winter. In late winter, it was about 4" high, and had 6-7 leaves. By mid April, it was 3 1/2 to 4 1/2 feet high, and showing heads, and I cut it with a string trimmer and dug it into the soil. I planted May 5.
Now winter rye adds organic matter, and holds nitrogen over winter (so it doesn't leach out), but it doesn't add nitrogen. There are legumes you can use as a cover crop, such as vetch, but they often need a longer time to grow, delaying your planting. They actually add nitrogen, but they don't have the tonnage per acre (or poundage per sq. foot, in a raised bed) that rye produces.
A couple of alternate options: plant a cover crop that winterkills (assuming you have adequate time to get it grown a bit before winterkill - you might actually provide the bed with a cover to try and extend the time of growth) which will protect the soil surface, suppress weeds and tie up nitrogen in decomposing the cover crop, but be less trouble to turn under in the spring than something that's actively growing. Buckwheat comes to mind.
Or, following that thought in a direction that requires less fall growing time, apply mulch, or compost (or simply compost materials to sheet-compost in place) and mulch heavily to the bed for more-or-less the same effect as a winterkilled cover crop lying on the surface.