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
The testnet Bitcoin wiki page references a "Testnet in a box" available here: sourceforge.net/projects/bitcoin/files/Bitcoin/testnet-in-a-box Jul 9, 2013 at 15:14
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)