So, putting 6 or more Mini Haniwa on your island will make sure a Kijimu will show up.
If you get a stray you don't want on your island, you can expel it so another stray can show up. If you haven't fed it yet, simply logging in and out of your island will get rid of it. Some stray pets are more rare than others, and may take a while to show up.
I've embedded a short video below to show how it works. You'll have to feed your pet each day to get it to 500 affinity until it changes from a "Stray" to a "Pet". Once you reach 500 or more affinity, you'll be given the option to give your pet a name. If you push "Cancel" or disconnect during this process, you'll have to wait until the next day to get your pet.
Read more: http://iruna-online.proboards.com/thread/46/pet-skills#ixzz3teoFss16