What is the best way to extend null check?

I tend to stick to the ubiquitous Guard class for this:

static class Guard
{
    public static void AgainstNulls(object parameter, string name = null)
    {
        if (parameter == null) 
            throw new ArgumentNullException(name ?? "guarded argument was null");

        Contract.EndContractBlock(); // If you use Code Contracts.
    }
}

Guard.AgainstNulls(parameter, "parameter");

And shy away from extending object, plus to the naked eye a method call on a null object seems nonsensical (although I know it is perfectly valid to have null method calls against extension methods).

As for which is best, I'd use neither. They both have infinite recursion. I'd also not bother guarding the message parameter, make it optionally null. Your first solution will also not support Nullable<T> types as the class constraint blocks it.

Our Guard class also has the Contract.EndContractBlock() call after it for when we decide to enable Code Contracts, as it fits the "if-then-throw" structure that is required.

This is also a perfect candidate for a PostSharp aspect.

I'd use internal static void ThrowIfNull<T>(this T o, string paramName) where T : class. I won't use internal static void ThrowIfNull(this object o, string paramName) because it might do boxing.