Why might a Christian be for euthanasia?

Simply put, because it reduces suffering. 'Love thy Neighbour' is key here. Jesus taught we should try and reduce a person's suffering. If someone is terminally ill and going through pain and suffering then some Christians would understand why it might be the 'lesser of two evils'.

Some other Christians, if not comfortable with active euthanasia, may be more comfortable with passive euthanasia. This is where a Dr may give drugs to a patient to alleviate pain, but it also speeds up the end of life. This is a secondary result from eliminating the pain.