Sorry in advance for the long read.
I'm 22 years old and just finished university for graphic design. I was approached by a small business which wanted me to design them a logo, stationery, and a website. I was only charging them $20/h NZD because it was under the table and my first client.
Anyway, I've finished everything. They're happy with the logo and stationery, but not the website. They think the whole website needs re-designing. They only told me this a month after I finished (once her husband started to get more involved with the deal). So I came up with another concept and sent them the wireframes for it, told them it would take roughly 50-60 hours. He sent a long email filled with analogies about flooring and how he wouldn't want his floor tiled twice, etc (it was an entertaining email). He left me with an ultimatum: do the re-design of the site for free or return the deposit he paid and walk away (after the hours already gone into it).
I would LOVE to hear from other freelancers and hear what they would do in this situation as I'm new to this and can't tell if I'm in the wrong or not. An important piece of information to note is that I didn't send them the wireframes of the site in the beginning (which was a mistake I now know) but does this mistake really have this much of a repercussion? And yes from now on I will use a contract.
If you read to here I would love your input, cheers.
Update: I took the advice from the answers below and told him I would need payment for a re-design or I could take it down and we part ways. After some discussion he said never mind and is now going to live with the current design and pay what he owes. (Sounds like he was trying take advantage and get extra work for free).