ARM was always much more common in mobiles, going all the way back to the old Newtons and first iPod, and it was very big in PocketPCs and cell phones long before they had the big app ecosystem they have today. Starting bigger they just had more momentum, and as a unified software platform became more important that dominance continued.
So I'm not sure it's that ARM overtook MIPS in the markets where it was big, but rather those markets got stomped out by other ones, like smart phones and tablets beating out PMPs and PocketPCs. You might be thinking PSP, but that was probably more influenced by Sony's past licensing experience on PS1 and PS2. And MIPS didn't migrate to ARM on consoles but to PPC. The reason for that is obvious: it was advantageous to move to IBM for manufacturing and the easiest solution from there was to use an IBM CPU core. Besides, there's nothing really competitive in the high end for MIPS anymore.
There's still some movement in mobiles for MIPS, like for instance Ingenic's SoCs. A lot of Chinese handhelds use them, like the Dingoo and the upcoming GCW will too.