Abelas means Sorrow, so whilst it is unlikely that a Dalish mother would deliberately give that ill omened name to her child, in the codex it is effectively a nickname that is given to him because nothing ever seem to go right for him. It makes sense to me that this sort of thing would happen among the Dalish, particularly in the transition from childhood to adulthood. So when you get your vallaslin, you might well be given a sort of adult title/name that you will then go by, particularly in your interactions with those outside of your clan. It is a bit like the Qun, but not the same since you do still have your own identity. It is also clear that taking different names during the course of your life was a practice of the ancient elves.
Abelas was not the original name of the character in the Temple of Mythal. Apparently he had a different name before he entered her service and then took a new name when he did. He only took on the name of Abelas after her death on assuming leadership of the survivors guarding the Well. This could also just easily be called "The Well of Abelas". When at the end Solas says he hopes he finds a new name, this is entirely consistent with this. In fact it would seem that Solas' actually words in translation are: "Your time as a Keeper/Guardian is at an end". So by implication he is saying that Abelas needs to find a new name by finding a new duty.