Do you believe that Greenland could become an Independent country one day and gain its independence from Denmark?

Yes, it's probably just a matter of time.

Why would the people want to, Denmark granted home rule to Greenland; in 2008,

Under the new structure, Greenland has gradually assumed responsibility for policing, the judicial system, company law, accounting, auditing, mineral resource activities, aviation, law of legal capacity, family law and succession law, aliens and border controls, the working environment, and financial regulation and supervision. 

The Danish government still retains control of monetary policy and foreign affairs including defence. It also provided an initial annual subsidy of DKK 3.4 billion, set to diminish gradually over time. 

With a population of just 56,000, who are in all those respects which affect their country independent, with the backing and protection of Denmark and through them the EU, I cannot see any possible reason why they would  want to change anything.

They may not want to be.  Do they have enough industry to generate the money it would take to be independent?

Hello, i add, an United nations members are unions or empires -horizontal democratic systems-, but not countries -not vertical political systems-, thus we shall think how to connect states but not to separate it, thus a Greenland  is a Denmark itself but not separated from it, but i think that a Greenland is natural -land or geography- name for a Denmark , where a Denmark is political name, but we consider the valid map for any place is the natural map but not the political map, where natural map is from an united nations, but political map is from the non valid enemy -the imperialism-, thus the true called it a Greenland place but not a Denmark place, but we consider unions or empires -horizontal systems- but not country, thus logically a Greenland is city from good Christians representation empire called -a Commonwealth of an an united nations subsystems-, but a Greenland not belong to an Europe Union, but an Europe Union is good too and from an united nations subsystems.

One suspects Greenland may be one of those places that actually benefits to some degree from the warming climate.