Re: What part could Russell's Paradox play in quantam mechanics?

Russell's Paradox is a mathematical aspect of set theory, see

http://plato.stanford.edu/entries/russell-paradox/

Russell's Paradox was not developed in a search for alternatives to the 
Heisenberg Uncertainty Principle, and there is nothing inherent in the 
statement of Russell's Paradox that would suggest it holds promise as such an 
alternative.  Though mathematics is very significant throughout physics, not 
every mathematical concept can be assumed to have significance in physics.

That said, you may be interested is another concept, the measurement problem,

http://plato.stanford.edu/entries/qt-measurement/

You can find at least one related response in the MadSci Archives using the 
search term

     measurement problem

and many other by searching different combinations of the terms

     Heisenberg uncertainty quantum

Thanks for your question.