It seems that you, her son, has acquired your mother's materials and has already invested at least some time thinking about this as you are now asking this question of what to do with it.
Since you say no one else in the family has interest, the obligation is on you. So it will depend how much time you wish to invest.
If nothing else, you should at least take a day of your time and go through her items.
First remove anything of sentimental value to you that you want to keep.
Then go through all the material, and ensure that they are all relevant with regards to her family research and remove anything that isn't. If the materials are organized into boxes with a purpose, keep them organized that way. If the items are all mixed up, then put like items together and organize them to save space and make all the material as compact as possible. If boxes are partially filled, use smaller boxes or folders to save space. Try to be left with a single large box with all the material that is clean and organized. That is a must. Nobody will want anything that is not clean or not organized.
It is possible that in a generation or two, someone in your own family will become interested in family history. Consider if there might be such a person. In that case, don't give the materials away. Just keep it in your basement or some other place you store your seldom used things. If you are storing it long term, put it in airtight plastic inside the box. Label it so you and others will know that is your mother's box of family history when they encounter it.
If you have children or grandchildren, siblings or nephews or nieces, then don't give the box away. Keep it and let them decide what to do with it when you're gone. Information about your mother and her (i.e. their) family will be much more important to them than anyone else.
What hopefully will happen through the years is that one day, one of your mother's descendants, or if she doesn't have any (other than you), then some close cousin becomes very interested in family history and you tell them about your mother's box and they'll be supremely excited and will take it from you. Ensure they are serious about it, because if they are, then they will use whatever they can from your mother's research and will document it in their genealogies and give credit to your mother for the pictures and information. They will likely also create online family trees at a service that other people can access and use as well.
Should you decide you don't want to keep it for what you might consider to be an unlikely case of a family genealogist sprouting, then your best bet might be a local genealogical society or a local library. Again, the material has to be clean and organized for anyone to take it.
If you still can't find anyone to take it, and you do want it preserved, then you'll have to pay a bit to do so. Find a professional genealogist who lives in your area who will be willing to be paid to do the work to find a proper home for your mother's materials. I would recommend looking through the Association of Professional Genealogists (APG) directory to find someone who can help you.