Short answer, yes.
Long answer, only a judge can order a person to act, and a judge would make efforts in accordance with the laws of the jurisdiction to satisfy the debt if such an avenue were feasible and would not unduly burden the debtor (something similar to cruel or unusual standards in criminal punishment).
The judge would not likely order a mortgage, since the judge cannot issue a mortgage, nor can he or she necessarily compel a third party to involve themselves in the current hypothetical fiscal debacle, but they absolutely could order someone to pay from excess income to satisfy a standard judgment. They could even go so far as to prohibit certain expenditures prior to satisfying the existing obligation (so that if you refinanced, you would have to satisfy the outstanding obligation prior to adding a new pool).
They could certainly also seize assets, though generally one's primary residence is exempt from seizure, excepting certain criminal circumstances, but failure to obey a court order could lead to seizure. In the case that the value sky-rocketed, they could force the sale because it would no longer fall under New York's exemption, and could be used to satisfy a debt, under the presumption that a new, suitable residence could be procured from the remainder of the proceeds of the sale.
Of course, I'm not a lawyer, I just play one on the internet. For specific legal advice see a licensed attorney in your jurisdiction, and verify that these kinds of generalities apply to more than classroom case studies.