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Actually, the thing that would be most appreciated - and valued - is a hand written letter on nice stationery giving congratulations and thanking him for his help in your own work.

Short, professional, sincere.

He will save it forever.

This is culturally dependent. You should follow local cultural norms.

When I was in Sweden, the PhD candidate would, after their successful PhD defence, organise a party, with typically between 30 and 50 people invited, such as family and friends from the office and outside. If they were supervising a few undergraduate students, they would probably invite them. At such parties, it was entirely common that people would bring gifts, which would all be placed on a table. If such a party exists and you are invited, by all means bring a small gift, similar to what you would bring if invited to a birthday party. I received food, books, souvenirs, and a few joke presents.

However, if such a party does not exist or you are not invited, and such gifts are unusual in your location, then it may be more appropriate to stick with a card such as suggested in other answers.

It is fairly typical for a student to give a thank you gift to their mentor when finishing. These gifts cause minor issues:

Is it ethical to accept small gifts from students?

Is it appropriate to buy a "thank you" gift for a PhD supervisor?

but you are not talking about a thank you gift, but rather more of a congratulations gift (e.g., like a wedding or baby gift). These are also awkward and the general advice is a group gift

Advisor's wife is having a baby, should we be getting him something?

Advisor getting married, should grad students chip in for gift?

Thinking about this in light of you having given a Christmas gift

Is it appropriate to give university lecturers Christmas cards?

makes me think this is all very weird. I would not want a congratulatory gift (especially one of any monetary value) from an individual student. If he is a mentor of a number of students, or works in a lab with other doctoral students, maybe you can take part in a group gift/card. I would avoid doing anything as an individual. This becomes especially true if you have not graduated yet.

Ask him for advice about buying wine. If you actually get advice, use it to select a nice bottle for him. If demurs about the wine advice, make a gift to the university in his name. They'll send a nice card and won't mention the amount.

One gift that is often really appreciated is a nice pen. Not a cheap one, but a fancy one, in a nice box. In some cultures it is the habit to not buy it yourself, but to wait until someone offers it to you. You can get them from 50-200USD.

https://upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Fountain_pen_writing_%28literacy%29.jpg/1200px-Fountain_pen_writing_%28literacy%29.jpg

I agree that it's culturally dependent, but in many cultures, you could bring sweets or something else that's edible, for after the thesis defense. I would personally consider chocolate from a higher-quality brand. It's not something of great value, but it's not something you'd just by at the corner kiosk; it is associated with some level of affection, but - unless you buy a heart-shaped box - it won't be construed as romantic. Most people (in my experience) like it, but at the same time it's acceptable not to open and have it right away if they don't want to.

As for a thank-you letter - that may be appropriate in general, but I don't see how that relates to that mentor's thesis defense in particular. Perhaps a congratulatory card which also has a couple of thank-you lines would go well with the chocolate though.

PS - Chocolate is also kind of a generic gift. It might well be better to get something which would be specifically appreciated by your mentor. But this answer is intended for other readers as well...

A nice matted and (optionally) framed photo of the city/town in which the university is located (e.g. from a local photographer) might be a nice gift, especially if it has an iconic skyline or downtown. This is even if the mentor doesn't plan to continue in the same city, because he still spent a number of years there and it had an impact on his life. It's also a nice relatively neutral item that can help decorate a future office.

In addition, +1 to Richard Erickson's answer about a thank-you/congrats card with a paragraph or few from you.