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House Cut Potato Fries

It's a name on menu, and what does it mean? Does it mean that it's sliced in the restaurant, or does it mean that it's grown/harvested in the restaurant or in it's farmer (not sourced from third-party)?

It means they used whole potatoes and made their own fries (French fries in the US or crisps in the UK, etc). The fries were not pre-cut and frozen. They were not bought by the restaurant in plastic bags.

House can refer to an eatery or bar. The other meanings of house cut are not relevant to this context though very interesting.

Fries suggests a North American locale; in other parts of the world, they may be known as chips or finger chips.

This is a sense of house as an establishment or venue, as Merriam-Webster defines it:

7a: a place of business or entertainment

a movie house

the local fish house

a house of prostitution

7b(1): a business organization

a publishing house

7b(2): a gambling establishment

7c: the audience in a theater or concert hall

So, house cut potato fries are potato fries which have been sliced at the restaurant (or hotel or club or whatever the establishment is), as opposed to being processed at a factory and shipped to the restaurant pre-cut. This is a way of marketing the dish as fresher and more carefully prepared, and are perhaps emphasized here because fries are closely associated with fast food or other low-quality dining.

Establishments which source food directly from farmers and other producers, as opposed to food processors and distributors, would most commonly refer to it as farm to table or farm-to-fork. These terms, and others like them like artisanal, are not well-defined and are not always meaningful—after all, even McDonald's food originated at some farm somewhere.

A possible alternative meaning is that the fries have been sliced in a particular way that is characteristic of the house, perhaps in a certain shape or pattern. You might see this in a food review, e.g. "The sandwich was served with fries in the house cut" or some such, but it is unlikely to be used in this way on a menu.

The same sense of house appears in other phrases. The same menu might offer a house wine (house red, house white, etc.), a standard wine that is always available. In contrast to house-prepared dishes, the house wine tends to be associated with mediocrity, since it is commonly purchased in bulk and sold at a low cost, but actual quality will vary. To avoid this connotation of house, establishments may present the offerings they consider representative as signature, e.g. signature cocktails, signature cuts, but this word is so overused that it has lost any real significance to most consumers.

Beyond food, House rules are variations on a standard game peculiar to a particular establishment. This is most commonly associated with gambling in card games, but for example, most people play the board game Monopoly with various non-standard rules that were never officially acknowledged until recently, and recognized by the publisher as house rules.

An organization may similarly enforce a house style, set of rules for writing, graphics, identity, and so forth that are peculiar to that organization. For example, the University of Colorado dictates that its published works follow the Associated Press Stylebook overall, but not strictly—it enforces various Exceptions to AP Style—and may be said to follow a house style.

Instead of House cut you could also say Home cut, it really just means that the potato fries are cut at the restaurant.

House in this context refers to the idea of something being produced/made on the site which it's being sold/offered.