Is there are any way to use bitcoin or alternative crypto-currency in local network? I have small home network without internet and have an idea to have a local currency to encourage the children.
You can set up a local Bitcoin testnet with the Bitcoin-Qt client and some relatively simple configuration. Note that you will need a miner to be processing transactions when the the testnet is in use. More information is available at: https://en.bitcoin.it/wiki/Testnet
If you are not tied to bitcoin you could also look at defining your own familly currency with an Open-Transactions server (advanced difficulty - https://github.com/FellowTraveler/Open-Transactions/wiki) or with Ripple (intermediate difficulty - https://ripple.com)