In theory anyone (including non tax payers) can gift aid, however as a non tax payer you would be liable for the extra amount claimed.
As others have said pick charities that you have some affiliation to, for instance I wouldn't expect an Atheist to be interested in giving to a church or someone with little interest in animals to give to a dog's home.
Depending on the organisation you may find that there's an element of political involvement, in that there will be campaigns that they would like you to be involved in (not that they support a political party). You can generally see what these are likely to include by looking at their website for current ones. This can result in two things, firstly it requires additional funding, however it can often result in better results than if they had spent the money on the symptoms.
As an example, if a charity can get the government to fix a policy which causes problems then that results in a better outcome than spending money on those who are worst impacted by the policy. Even though that means that they have higher staff costs.