It is not likely that this is mononucleosis. That infection is not seen in this way. It is possible that an infection could be coming from the sinuses or another area, such as the tonsils themselves. If this is an infection, no home remedy is likely to get rid of it. An ENT specialist is still the best person to evaluate the problem. This could include culturing of the sinuses and tonsils for a variety of infections, including fungal infections.
These are probably tonsil stones; he could consult an ENT.
More info:
http://www.doctorhoffman.com/xlith.htm
Tonsilloliths
These things are called tonsilloliths ("tonsil stones.") You have described them perfectly. They are typically white, or whitish-yellow, irregularly-shaped, foul-smelling, and usually smaller than a pea. The tissue lining the tonsils (oral mucosa) is fairly sensitive, so people with this problem often report that they feel an irritation in the back of their throat and are able to pick these critters out of their tonsils.
Like your skin, oral mucosa sheds continuously. The dead cells become incorporated in your saliva and then you swallow them. Yum! One more anatomical fact that you need to know: the surface of your tonsils is pockmarked with deep pits, appropriately called "crypts" (since dead stuff accumulates in these pits.) Oral mucosa lines the crypts. Under normal circumstances, as this mucosa sheds, the dead cells leave the crypts and are swallowed. In some unlucky people, like you and your sister, the dead cells accumulate and glom together to form hard little balls. All of this dead stuff makes great food for bacteria, and of course your oral cavity is colonized by all kinds of bacteria. Consequently, the tonsilloliths are ripe with bacteria. This accounts for the smell.
The only sure-fire method for ridding yourself of this problem is to have your tonsils removed (tonsillectomy), but there are nonsurgical alternatives which you should certainly pursue before you ask an ENT to take out your tonsils: