Answer: The answer really depends on how much sun the area receives. If it receives sun all morning through noon (about six hours of sun) then it would actually be considered a full sun location. If it receives four hours or so of direct sun then it is partly sunny and with less than that, it is considered essentially shaded. I can't think of a weeping shrub for shade. In an area six by six, you do not have room for even the smallest type of tree. You would be looking at the most for a mid sized shrub.
If it is partly sunny you might try one of the newer hydrangea cultivars that bloom on both old and new wood such as Endless Summer. This should bloom from late summer until frost and provide an explosion of color. Or, in partial sun, you could try the PeeGee hydrangea (Hydrangea paniculata). This Hydrangea grows to the size of a small tree and seems to nearly weep when it blooms because the large white flowers pull the branches down. Or, you could plant a clematis vine on a trellis.
In a truly shaded spot, I would suggest you try a slightly different approach. You could use a very large weather proof urn or planter and fill it in summer with colorful impatiens surrounding a large leafed hosta and possibly a trailing variegated ivy planted to one side. This would look dramatic in summer and sculptural in winter.
I hope this gives you some ideas.
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