Why did Palestinians elect Hamas if most view them as a militant organisation?

This is a common trait in a state of conflict where people naturally choose the hard-lined leader in order to protect their people with strength and courage against an external threat, rather than with their PhDs in diplomacy.

We saw the same scenario with the rise of a General who became Prime Minister in Ariel Sharon, or Binyamin Netanyahu who was part of the General forces unit and a captain, and even Menachim Begin who was part of the Irgun terrorist group that bombed several British buildings and traded with illegal weaponry.

This is even evident between two military leaders who took separate paths in their methods of leadership regarding the zealous persona of Gamal Abdel-Nasser who was popular in Egypt because he promoted protection against the Zionist invasion, rather than the more diplomatic approach to peace in Anwar Sadat, who subsequently was assassinated for his attempts at peace. This similarity can also be seen in Israel when Yitzhak Rabin was also assassinated for his attempts at peace in a heightened time of conflict.

This stems from the propaganda created for warfare. The classic situation where newspapers and radios promote the perception of "near-annihilation" as troops battle to defend their country's existence. While a people face the depressing realities of daily bombardment, they need a hero to turn to in order to give them hope of victory.
In the first world countries today, we witness people naturally inclined to seek justice in internal affairs rather than in warfare against an external entity. This is especially common amongst the youth who have lived all their lives in a world without any direct imminent threat from external entities, as was the case before their forefathers fought to win their own independence.

Gaza is no different. Rather than the people seeking a government that can provide them financial stability, economic opportunities, and social justice, they believe that none of this can be achieved with the constant humiliation of occupation and a disruption of normality with ongoing checkpoints and land confiscations in areas that were replaced with settlements, as was the case before the seige began in Gaza in 2006. With absence of peace in their everyday lives and the uncertainty of their future, those who become popular were the highest bidders for security.

Hamas' militant approach is the strongest and most secure option for the people of Gaza because no one else can provide them with the protection they need. It is unfortunate that since they have taken office their militant approach has been like punching the apartheid wall with their bare fists; achieving nothing but reactions from those who imposed the blockade around the Strip, rather than a stable solution for the people's livelihoods. These reactions by the occupying powers continually result in civilian casualties even after democracy was implemented in the region with Hamas. That particular democracy was shunned, and was still frowned upon even after Hamas were willing to accept a truce with Israel based on the pre-67 borders in the same year they were elected into office.

What resulted from the elections that birthed Hamas' political careers was evidence that violence ultimately spawns violence, just as the Gaza siege in 2008/2009 will undoubtedly bring up another child who witnessed brutality with the horrific scenes of burned bodies and the pools of blood. Those scenes could only traumatize a collective group of youth that will grow up in a violent environment and a vengeful disposition.

Israel's mistake in dealing with the situation of trying to remove Hamas has not yet been realised. Rather than a collective punishment imposed on Gaza because the people elected the "wrong leaders", the withdrawal of violence will actually have a positive effect, and it will naturally dissolve support in such reactionary groups amongst popular Palestinian opinion.