I don't have a machine which runs 24/7, so server is not an option. Besides, I am getting IP addresses from DHCP running on home router provided by my ISP. I've read a bit about Avahi. That thing is supposed to do several things:
1. Helps discover services on LAN, link finding shared printers
2. When a network connection is being initiated, it sends multicast query, like "who is host named 'Lexx'". When answer arrives all host on LAN which run Avahi can store it in their caches
3. Provides additional nsswitch.conf interface for "old school" C runtime library functions, like 'gethostbyname()', allowing 'normal' programs, like ping to benefit from it.
I was able to resolve names by avahi-resolve program. My guess is it means that multicast part works. However, I can't ping anyone. Also, libnsssmdns is missing, so I figured out that nsswitch interface is not working.