There are several ways in which a site can gather funding using bitcoins.
First of all, this is exremely easy to use, because no banks are involved. Just a bunch of python scripts. Or a regular bitcoin client.
1) you can let people send donations with bitcoins, but since bitcoin is not too popular yet, that will not be big income, until it gets popular.
2) so make it popular. For example implement a bitcoin-based forum system, in which people can choose to participate. If they choose to participate it works following:
2.1) users pay a tiny amount, like 0.001 BTC to post a comment.
2.1.1) If someone who is participating liked their post, they can choose an amount of BTC they want to tip the poster
2.1.2) if someone replied to it, the poster gets 80% of those 0.001 that the replying perspn paid in order to post
2.2) forum users who did not choose to participate, can post comments for free, but also cannot earn any BTC for being a helpful community member.
the remaining 20% for each posted comment goes to the website.
Now bitcoins get more popular on the website, and the website can earn money without advertisments at all.