Your query will result in populating some object  (data source) with data
and the ddl will databind to that object. You will need to persist the data
source between postbacks. You can do it in a session variable. Or you may
choose running the query on every postback to re-populate the data source.
Once you know the selected id, you can search your data source to locate the
data item and get the city from the data item.

Several ways...

1) Unless your DataSet is quite large, you can store it in a client-side
JavaScript array - this would allow you to retrieve the associated city
client-side without any postback / callback...

2) Concatenate the ID and city with a separator charactor and use that as
the value of the dropdown e.g.

   1¬London
   2¬Paris
   3¬New York

Then, when you need to, you can simply split the value on your character
separator e.g.

MyDropDown.SelectedValue.Split('¬')[1];

3) Use AJAX to either look the data up from the database as and when
required or, (maybe) more efficiently, persist the data in Session or
Application cache if it's unlikely to change very often...

I'd go for the third option, personally...

string stateid = StateList.SelectedItem.Value;
string sql = "SELECT cityid, city FROM cities WHERE stateid='" +
stateid + "'";