// Construct a JObject.
var jObject = JObject.Parse("{ SomeName: \"Some value\" }");

// Deserialize the object into an ExpandoObject (don't use object, because you will get a JObject).
var payload = JsonConvert.DeserializeObject(jObject.ToString());

// Now you can serialize the object using any serializer settings you like.
var json = JsonConvert.SerializeObject(payload, new JsonSerializerSettings
{
    ContractResolver = new DefaultContractResolver
    {
        NamingStrategy = new CamelCaseNamingStrategy
        {
            // Important! Make sure to set this to true, since an ExpandoObject is like a dictionary.
            ProcessDictionaryKeys = true,
        }
    }
}
);

Console.WriteLine(json); // Outputs: {"someName":"Some value"}