I haven't done it in years, but I can recall putting some small rocks and metal nuts/bolts inside the feet of some of my early papier mache sculptures in order to give the feet the weight they needed to make the figure stand unassisted.  These days, if one of my figures can't stand by itself, I usually super glue it to a transparent plastic base (the next best thing to invisibility is transparency).

You can papier mache over just about anything, so I imagine you could do it with blu tack--I don't know if blu tack ever hardens though, so you might end up with a "squishy" spot there if you don't layer enough papier mache over it.  Any clay that air dries or a self-fixing chemical epoxy should serve equally well.  If it air dries, I'd give it the necessary curing time before applying any PM over it.

Alternatively, as you're still at the armature stage, you could try readjusting the pose of your figure in order to get the center of gravity such that it can stand on its own without weighing the feet down.

I've used plaster of paris to give weight to my candle sticks,it woks really well. I always keep it handy.....as it is quite cheap also.

A stone of a suitable size, held into the structure with masking tape or superglue can weight the project very easily.