A phoneme is the smallest contrastive unit in the sound system of a language.
First, “Definitions of phoneme” in the context of a spoken language is not unanimous. SIL's definition above is good, but if the language has no sound system, do like Steven says, find the contrastive units. But there are problems.
Here are two ASL signs formed by putting a pointer finger on the forehead and moving it sideways, where bending the finger at the end changes the meaning. Each sign consists of an initial position, a movement and a final position. The contrastive feature is the +/- bent finger in the final segment.
The modern notion of a phoneme is a temporal segment, so one can't say the finger, or just the handshape is a phoneme, we have to include the entire feature bundle: orientation;palm down, location;forehead, and everything else that happens simultaneously in that segment, would all be included as one phoneme. With around a hundred contrastive handshapes, and the same number of locations, plus orientation, movement and facial expressions, there would be literally millions of such “phonemes”, and this sort of defeats the purpose of having them.
If one takes an older, Bloomfieldian definition of “phoneme” any contrastive element counts (e.g. Chinese tones), so just the bent index finger alone would be a “phoneme”. But that's even worse!
Mostly, phoneme theory has nothing to offer sign linguistics, and every model of signed structure is feature-based and uses autosegmental or other non-linear phonology. Upshot is, no phonemes in signed languages.
Well there aren't phonemes in signed languages because phonemes are specifically sounds. But if you're asking whether signed languages have an equivalent of phonemes then indeed they do, and they are called cheremes.
Then again the Wikipedia article on cherology, while a bit thin, actually does seem to say that the term phoneme was formerly used even in the context of signed languages prior to 1960 when the terms were coined.
Even as linguists we can forget that meanings can drift away from etymologies (-:
No sooner had I mentioned the drift than user mmoosman pointed out that cherology and chereme have fallen out of favour once more with phonology and phoneme now covering both spoken and signed languages once more.
Is there a specific question about whether a particular candidate can be considered a phoneme in sign language? Perhaps if you frame your question in terms of the research you've already done on the field (and it doesn't take much to start), people can build on that. That way, you can take advantage of the diverse array of backgrounds here to help you arrive at new knowledge. I feel the way the question is phrased now, it might be banking entirely on one of the 50-or-so private beta-ists on knowing sign language phonology, which might be unlikely.
That being said, what a phoneme is is no different in sign language than in natural language. The fact that word phoneme is derived from a root meaning sound doesn't affect the more abstract meaning that it's taken in academic linguistics. It's the underlying representation of a minimally contrastive unit. If changing the location of a gesture and nothing else changes the semantic meaning produced (in ASL, and all other full sign languages as far as I know), then each location (or some more abstract construct that includes or influences location) is mutually contrastive. If changing the hand shape and unilaterally change the semantic meaning, then each of those shapes are mutually contrastive.
Once we know all the mutually contrastive locations, movements, and hand shapes (and other things), we discover relationships between them and rely on native speaker intuition to wean this down into a candidate list of phonemes, whose compositions can deterministically produce every word in the lexicon.
Historically, signed phonemes were considered to fall into 4 (or 5 depending on who you ask) categories that are sometimes called parameters. These are handshape, location, movement, palm orientation, and non-manual markers. Interestingly, every sign must have at least one of each of these parameters. The parameters are made up of features (handshape includes selected fingers, abduction, flexion etc.). This video shows some examples of minimal pairs of each parameter. The parameters are useful to an extent, but are somewhat outdated. Newer theories recognize that there is an awful lot of simultaneity in sign language (a single handshape can span multiple locations, for example), and have begun appealing to an autosegmental perspective. This is similar to the treatment of tone in tonal languages like Chinese where a single tone can span several segments. I wouldn't go so far as to say that there are no sign phonemes, but the degree of non-linearity in sign language certainly does challenge traditional ideas about the phoneme.
I'm not an expert (or even a linguist), and my knowledge on this point is at least 10 years out of date (learning about ASL linguistics in high school from my dad who's an ASL interpreter). But one system for phonemes in ASL is based on location, movement, and handshape. For example, the "A" handshape can be done either with the base of the thumb touching the side of the hand or with the thumb sticking further out, and either is the same phoneme. Whereas, if you put the thumb in front of the hand (the "S" handshape) that's a different phoneme. Similarly, the difference in location between upper arm and lower arm is phonemic, but the exact location on the upper arm is not. Wikipedia has some more detail.
Phonemes are the minimal contrastive linguistic unit along the syntagmatic axis; think of a string of units listed sequentially. Phonemes themselves are comprised of features, and this is within the paradigmatic axis; think of one position on the syntagmatic axis and see what it is like on the inside. Sign language phonology has not yet availed itself of this distinction with regard to the sign language phoneme.