The question is: Can I initialize var with null or some empty value in C#?

C# is a strictly/strongly typed language. var was introduced for compile-time type-binding for anonymous types yet you can use var for primitive and custom types that are already known at design time. At run time there’s nothing like var, it is replaced by an actual type that is either a reference type or value type.

When you say,

var x = null; 

the compiler cannot resolve this because there’s no type bound to null. You can make it like this.

string y = null;
var x = y;

This will work because now x can know its type at compile time that is string in this case.

Well, as others have stated, ambiguity in type is the issue. So the answer is no, C# doesn’t let that happen because it’s a strongly typed language, and it deals only with compile time known types. The compiler could have been designed to infer it as of type object, but the designers chose to avoid the extra complexity (in C# null has no type). One alternative is

var foo = new { }; //anonymous type

Again note that you’re initializing to a compile time known type, and at the end its not null, but anonymous object. It’s only a few lines shorter than new object(). You can only reassign the anonymous type to foo in this one case, which may or may not be desirable.
Initializing to null with type not being known is out of question. Unless you’re using dynamic.

dynamic foo = null;
//or
var foo = (dynamic)null; //overkill

Of course it is pretty useless, unless you want to reassign values to foo variable. You lose intellisense support as well in Visual Studio. Lastly, as others have answered, you can have a specific type declared by casting;

var foo = (T)null;

So your options are:

//initializes to non-null; I like it; cant be reassigned a value of any type
var foo = new { }; 

//initializes to non-null; can be reassigned a value of any type
var foo = new object();

//initializes to null; dangerous and finds least use; can be reassigned a value of any type
dynamic foo = null;
var foo = (dynamic)null;

//initializes to null; more conventional; can be reassigned a value of any type
object foo = null;

//initializes to null; cannot be reassigned a value of any type
var foo = (T)null;