Which one is more grammatically correct as a shop name, Valhalla’s Dojo or Valhalla Dojo? It’s a name for a game centre. My partners and I can’t seem to agree on this.
Both are grammatically correct. They just mean different things.
In this case, Valhalla is taking a genitive role. This is the dojo that belongs to, is part of, or emanates from Valhalla. It's commonly referred to as a possessive, indicating that Valhalla possesses the dojo, though it's a bit more complex and has a wider range of meaning than that would suggest.
In this case, Valhalla is possibly taking an attributive role. It is an "attributive noun", used to indicate something about the attributes of the other noun to which it is attached. It is showing some association between the dojo and the idea of Valhalla.
However, it could also be said to be taking a purely nominative role, the name of the place being Valhalla and it being a dojo.
Of course, if the thing being so named isn't actually a dojo, the implications may be quite different.
Either is fine in it's own way. Both are commonly used in business names.
Means 'the dojo of Valhalla', or 'the dojo belonging to Valhalla'. That makes grammatical sense, however unlikely one might feel a dojo in Valhalla would be. Businesses do often use this device to associate a business with something else.
Means 'the dojo called Valhalla'. The dojo doesnt necessarily have anything to do with Valhalla, it just happens to be named after it. Think of it like 'Pepsi Cola' ... a cola called Pepsi. Pepsi doesn't inherently mean anything other than a way to distinguish itself from other colas. Nevertheless, by using the name Valhalla you would be creating an association in the client's mind.
So different cuisines will often have restaurants called things like 'Bombay Gate' or 'Peking Castle', but it would also be fine to say 'Bombay's Gate' (or 'Gate of Bombay') or 'Peking's Castle'.
fred2's answer is correct, but it is worth going a little further. A construction like "Valhalla's Dojo" of course uses the apostrophe as a possessive marker. If Valhalla were a person it would be universal to use the apostrophe, and an error to omit it. It is less common to use the apostrophe attached to a geographic location or building, because the connection between the words is an association (the dojo that is at/in Valhalla) rather than possession (the dojo belonging to Valhalla), so "Valhalla Dojo" would be a more usual construction.
An exception is when the location is used as a proxy for people who have a strong connection there - typically in the context of national or local pride. So "America's National Parks" or "Italy's coastline" would be common usages which carry some flavor of a sense of "ownership" by their peoples.
Another common use of the apostrophe is explanatory. For example, it would be usual to refer to "the Tokyo tower", but if you were trying to specify one out of several items, you might refer to "the Tower of London's east gate".
Using Valhalla's Dojo is not wrong - but omitting the "'s" is almost certainly better.