What is the basis for the argument that Jesus DID have the ability to give in to temptation? I understand that some people believe that Jesus actually had the capacity to sin, that is, to give in to the temptations that He suffered.  This seems to suggest that God has the power to sin.

In a sense, then, the redemption of mankind hung in the balance, so to speak, during the temptations of Christ.

What are the theological arguments in support of this position?
The argument comes both from the fact that Jesus was fully human, and from the Hebrews 4:15, which says (in the King James)


  For we have not an high priest which cannot be touched with the
  feeling of our infirmities; but was in all points tempted like as we
  are, yet without sin.


This outlines the fact that even though Jesus is fully God, He was also fully human.  He felt all our infirmities and weaknesses, and yet, as God overcame them.

A complete theological statement would be "because Jesus was fully Human, He had the capacity to sin, but because of His perfect nature, also being fully God and therefore perfectly good, He could not sin."  

Both are true, but people have a hard time wrapping their head around that, choose half the truth, and fall into error.