C# referring to object variable with string

Dictionary is not part of the language & is provided in the framework.
Use Dictionary<Key, Value>.

e.g.

Dictionary<string, string> myData = new Dictionary<string, string>();

myData.Add("first", "stack");
myData.Add("second", "overflow");

Console.WriteLine(myData["first"]);

It depends.

If you genuinely mean an object you want:

objectName.variableName = "Some value"; 

If you want a Map (also known as an associative array) your best solution is a Dictionary:

Dictionary<string, string> d = new Dictionary<string, string>()
d["variableName"] = "Some value";

You could use a dictionary...

using System.Collections.Generic;

public class x{
  public method1() {
    var objectName = new Dictionary<string, object>();
    objectName["variableName"] = "Some value";
  }
}

Or, use strongly-typed properties (safer and fast running), recommended where you know variable names at compile-time.

public class Person{
  public string Name {get;set;}
  public int Age {get;set;}
}

// and use it as follows in your functions

var person1 = new Person() {
  Name = "Fred",
  Age = 21,
};

// again, to demonstrate different syntax to do same thing
var person2 = new Person();
person2.Name = "Danny";
person2.Age = 2;
person2.Age = "x";  // won't compile - expects int, hence safer

What type of variable? It sounds like a dictionary/hashtable is most suited. These have index operators [], so the syntax will be identical.

C# doesn't have "objects with variables". C# objects have properties which are accessed like MyObject.PropertyName.

What you're referring to is a key-value collection. A Dictionary is C# implementation of this. Note that a dictionary only allows for unique keys. You can use it generically so you can use it like this.

Dictionary<string, string> myValues = new Dictionary<string, string>();
myValues.Add("variableName", "variableValue");

or

Dictionary<string, int> familySizes = new Dictionary<string, int>();
familySizes.Add("simpsons", 5);

As you can see you can choose what datatypes you use.