How do I secure a one-armed toilet paper holder, which constantly slips off the mount?
This looks like a bad piece of engineering design to use a slip finger bracket to mount a cantilever paper holder. This type of bracket would work pretty good for a towel bar that was mounted on both ends or possibly for a soap dish that was mounted right on center. 

I can suggest two possible courses of action that would solve the problem as long as the finger bracket on the wall was very nice and secure. 


You could mix up some good quality epoxy (the kind that takes 24
hours to set up) and force it into the finger slots on the bracket
and on the fingers of the wall bracket. Then install the roll hanger
and let the epoxy set up. Note that this is a pretty permanent fix
and future removal would require cutting the hanger off with a dremel
tool or some such.
Alternatively you could clean up the wall and hanger assembly to
remove any oil and soap film and then re-install the hanger. Rubbing
alcohol can work great for this. Then run    a small bead of clear
silicon sealer all around the perimeter of the bracket. This
sealer will take the twisting load off the fingers and transfer it to
the tile which should stop your problem. For removal the silicon
sealer can be cut away with a razor blade.


If it is the finger bracket on the wall that comes loose then you'll have to investigate how it is currently mounted. If the two screws there are already go directly into a stud or wooden cross member then some longer screws may be in order. If the screws just go into some cheap plastic hollow wall anchors then you may want to investigate some more secure types of metal anchors such as toggle bolts or molly screws.