What is the semantic difference between "select of your own choosing a partner" and "select a partner of your own choosing"? Slightly awkward grammar aside, the first seems to suggest that the choice is whether or not to participate in the selection process, while the second seems to suggest that the choice is of the partner, but is there any real difference between these two wordings?
The first form sounds decidedly odd to me, but I can't see it meaning anything different to the second.
They both seem tautological anyway. Select means choose; I don't see the point in saying Choose a partner of your own choosing. I think just Choose a partner is simpler and better.
To convey that even making a choice is optional, say You may choose a partner [if you wish]