If a person rejects all organized religious beliefs involving the Judeo-Christian God and any other religions that worship a god, but still believes (without much doubt) that a higher power does exist. What religion or belief system would that be classified as?
This is as much a linguistic question as philosophical so I'll put in my two cents. It hinges on just what kind of a "higher power" she's talking about, so I would urge you to get more details about that.
A bare consensus of definitions of the word "religion" requires belief in one or more gods; a god is a conscious, willful, supernatural creature who rules over some or all of the events in the natural universe. A supreme being as it were, with or without capitalization, depending on whether there's one or more than one. If your mother believes in one or more gods, then she is a
theist.
If her "higher power" is not conscious or not willful, or in some other way is not comfortably close to being a human-like creature with supernatural powers, then if it does have power over our lives and the events in the natural universe, it must still be a supernatural force or entity of some kind. This would disqualify her from being a
theist, but she is still a
supernaturalist.
That's the best I can do. Various oriental traditions postulate supernatural forces but not, specifically, gods. The dao, for example. And Buddhism; it comes in many flavors but I think the majority of Buddhists do not believe in gods. In my opinion this disqualifies mainstream Buddhism from being called a religion.
The name animism is applied to a couple of different belief systems, but one of them ascribes souls to what we consider
inanimate--literally "soulless"--objects. That would make her an animist but I just don't think that's where you're heading with this.
As I said at the beginning, you need to find out more about what your mother actually does believe in, rather than what she does not.
At first I thought agnostic, but this person doesn't really have any uncertainty about the existence of a "higher power".
An agnostic is uncertain about the existence of gods, specifically, although this word has been used in non-standard ways.
They believe that it does exist, but just not attached to any, what they refer to as, religious "mumbo-jumbo". Does anyone know?
Well she doesn't get to decide what is and is not religious; let the dictionary do that. As for "mumbo-jumbo," that's in the eye of the beholder, and some of us regard all assertions of the existence of supernatural forces as 100% pure first-pressed virgin mumbo-jumbo.
You have to get her to tell you just what sort of "higher power" she's talking about. The natural laws of the universe can easily be regarded as the ultimate "higher power," and we have a name for the people who believe in them: "scientists."
If it turns out that she truly cannot articulate what she believes in, she may be an apostate who turned her back on the religion of her childhood because of some really awful experience in the religious community; yet she retains the yearning for the comfort of that certainty without retaining the object of the certainty. In that case I don't know what I'd call her except "unfortunate."
