Im not sure how to actually start an online game. I have tried starting a private game with my mother, but it doesnt seem to work when i search for her ip address.

http://www.gog.com/forum/heroes_of_might_and_magic_series/homm3_online_play

First off, to play this online and moreover over LAN - you dont need anything to download. Just clear GOG\GOG Galaxy version. Even GameRanger is not needed since you dont want to find someone to play online but already have opponent in LAN network.
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You put wrong IP then. If you at real LAN (connected by same Wi-Fi router or wired), ones who connect to hoster should put IP address starting from 192. and anded on specific number of hoster IP (.1 if for router itself - else number are same). Also some port after ":", but that work differ and mostly not required. Not sure about HeroesMM3.

You can use GameRanger also (online but wrap LAN so ingame IP 192.168... same as for LAN) or online hosting+connection with online IP address.

Ports usually opened automatically if no restrictions set to do this. Disable any possible protecting software\services.
Both of you should install GOG Galaxy and\or games to - strictly! - X:\Games\*game-name*\ and start game exe as "admin run". This is also disable Windows Defender for that exe (but not folder, this is why it should be not on C and only in Games folder)(ofc not disable else antivirus'es - turn them completely off - protecting and even self-protecting for some).

You dont need disable Defender (just dont set max strict protecting and\or add Games to exceptions). But disable Windows DEP and Windows Firewall.
Also, make sure you have DirectX 9 installed and updated (updates by dxwebsetup.exe + W.Update). Go to Control Panel - Windwos features on\off - Legacy - Legacy DirectX DiectPlay -> on and restart PCs.

If you use one Wi-Fi home netwrok both PC connection should be set as "Home\Trusted".

Do not put anything to executable parametres or put same as hoster.

Verify game files in Galaxy - at least twice and start it to main menu then exit and restart again. if you use GOG offline installer - reinstall it. Or you can add this build to Galaxy as Galaxy build, but do verify again.