Small, cucumber plant dying

It is certainly dying and there is little you can do now. It won't have been the bbq that affected it. Where have you been keeping the pot, looks to me as if it has been stretching for light and the soil/compost looks pretty damp to.

I wondered if it got a frost, was it outside?
but it is beyond saving. sow another seed, its not too late.

Just wondering if its a victim of "damping off" - that's a fungal disease that sometimes affects very young seedlings. it can result from dirty pots or affected compost, though your pot looks quite clean. Or maybe your compost was a bit too damp. If it was outside it could have been affected by frost - there was quite a severe one here a couple of nights ago.

Very fine-textured compost. Is there any garden soil in it? In pots, garden soil can cause problems with drainage and fungus, including damping off.