This action undertaken by the Palestinian Liberation Organisation at the UN is not taken in the context of strategy, but because it has stumbled into it and is trying to reclaim some political ground. I think the bid is taking place in a strategic vacuum, and therefore my analysis of what might happen at the UN is based on this being a consequence of political frustration and anxiety, rather than intentionality.
 
The move is not something that is born of a trenchant critique and reassessment of a failed way of going about advancing their causes. Going to the UN would make more sense as step one in a multi-pronged strategy to bring about a national achievement.

Despite an absence of clear intentionality, if Palestinians do proceed, just winning a vote in the General Assembly, then it gives them a little more recourse to international organisations. None of this is automatic.

The only viable Palestinian path to full UN membership is via the Security Council, and that route is blocked by the certainty of a US veto. Failure at the Security Council may itself be a drawn-out process. Any application would almost certainly have to be considered by a technical committee of the whole and that could take time.

Even if they go to the Security Council, the Palestinians would then deny themselves the option of going for a win at the General Assembly during this window of heightened UN attention. They might otherwise find their entire UN moment sidestepped by extended committee deliberation.