There are two words or phrases that are bandied about when euthanasia is brought up in a positive light: "compassion" and "putting someone out of his/her misery". One is wrong (and dreadfully so) and one is correct (and dreadfully so).
It is not "compassionate" to euthanize someone; at least, not according to what "compassion" means. It comes from the Latin com- ("with") passio -> patior ("suffer, endure"). To be compassionate toward someone is to suffer with them, not to remove their suffering (and their life along with it). Euthanasia is utterly anti-compassionate.
But euthanizing someone certainly does "put him out of his misery." Again, we must consider what "misery" and "miserable" really mean. The Latin root is miser ("pitiable"): misereo means "to show pity" and miserabilis means "worthy of pity." To show pity to someone means to show mercy to them. Indeed, the word "mercy" comes from the Latin misericors (misereo + cor "heart"), essentially meaning "a heart that shows pity". So yes, euthanizing a man puts him out of his misery! It removes from him the need to be shown mercy to; it removes from him that which compels another who has a heart to show him mercy.