If I have a little child with an Ipod (which is muktzeh) in its hand can I grab the hand and shake it out of its hand into a safe place to prevent the child from breaking it?
The Kitzur Shulchan Aruch (88:13) says that moving Muktza indirectly (as you described) is only allowed if you need the object you're shaking, and not for the sake of the Muktza object.
So you shouldn't be allowed to shake the kid in order to drop the Muktza, if your intention is only to safeguard the Muktza.
However, since you are not allowed to pick up a kid who is holding Muktza (ibid 14), you could decide you want to pick up the kid and cuddle/play; as a result you now could shake the Muktza off them (onto a safe place - ibid 13) in order to pick up the kid.
Disclaimer: I'm not a posek! Consult your Rabbi before accepting my opinion!
But to my mind, I think you could take the ipod directly out of the child's hand and put it somewhere safe.
Reason: a k'li shem'lachto l'isur (object whose main purpose is a forbidden activity on shabbos) can be moved "michama l'tzel" (from sun to shade) in order to protect it from being damaged. You don't even need to use a shinui - you can just pick it up and move it.
Likewise here - the ipod is in danger of being damaged; you can just pick it up and put it somewhere out of the kid's reach.
(I have deliberately left my previous answer "struck out", because while I'm embarrassed by my previous show of ignorance, I think it's a valuable lesson in consulting your sources before opening your mouth... thanks to @msh210 for questioning me!)
Turns out "michama l'tzel" is a specifically stringent case; you can't even do that with a k'li shem'lachto l'isur (one of the most lenient cases of muktzeh) (Shabbos 123b, Rashi ad loc. d"h ומחמה לצל).
There is, however, an eitzah (recommended "trick") for someone who wants to move something muktzeh in order to protect it from damage: invent some reason why you need to use the space it's in, e.g. if it's on the grass, you can decide you want to sit down in that spot specifically. Then you can pick up the object - and once it's already in your hands, you can go and put it down wherever you want. Even though it's a bit of trickery, it is permitted to do this for a כלי שמלאכתו לאיסור (Shulchan Aruch HaRav, 308:12).
But after all that, I'm not sure how that translates to the kid with the ipod - maybe you want to use his hand to put some candies in...? :)