The Only One I Ever Wanted

"Always" is usually used instead, but there is nothing wrong at all with using "ever" (aside from sounding a bit stiff).

I think part of the reason it sounds awkward is because of its logic. "The one I ever wanted" could refer to several things, especially if you've ever wanted a lot of things in the past. Instead, "the only one I ever wanted" can only refer to one thing and informs the listener that you have never wanted anything except for that one thing.

I would explain the usage of "only" in this phrase as being part of an old expression. It sounds right because this is the way the expression has been for a long time, and it goes back to a time when "ever" was probably not so stiff, as IK correctly points out it sounds today, when replacing the more normal sounding "always". If you say the expression without "only," the well-used expression is now just a phrase you have made up. This phrase still has meaning. But since it is not part of that expression, it now sounds stiff and leaves the listener to wonder why you didn't say always. Perhaps in a poem it would appear more acceptable, since we often have an interesting turn of phrase or hearken back to earlier days in poetry.

I would add that "only" in the expression adds some meaning. Since human beings are generally thought to be expansive in our desires, saying "the one I ever wanted," while indicating that you did in fact want that one and have always wanted that one, does not preclude the possibility that there have been others that you wanted. It comes close to precluding that possibility, but saying "only" really puts the issue to rest.