I shall take it as a compliment that I've been confused with Mushroom25 (who, along with Dzug2, is one of AB's recognised travel experts)!
The Hahn Airport website shows all bus links to the airport:
http://www.hahn-airport.de/default.aspx?menu=by_bus&cc=en
none of which are particularly useful. (i.e. there's no direct service from Stuttgart). There's also no train station close to Hahn.
Despite Ryanair referring to the airport as 'Frankfurt Hahn', it's a 1h 45m bus ride away from Hahn. Mainz is closer but it's still 1h 10m away. However there's a daily bus at 0350 from Mainz Railway station, which gets to Hahn Airport at 0700. On a random date that I've checked, there's a direct train from Stuttgart to Mainz at 2151 (arriving 2330) or a 2305 service, connecting at Frankfurt, arriving at 0044. (NB: There's only 4 minutes to change trains!).
You can search for trains between Stuttgart Hbf and Mainz Hbf here:
http://www.bahn.com/i/view/GBR/en/index.shtml
Of course, if you were going to change trains at Frankfurt Hbf anyway, you could also take the bus from there. The 2305 train arrives at 0042 (assuming, of course, that it runs on the day that you want - check as above!). There are daily buses to Hahn airport at 0300 and 0345, arriving at 0445 and 0530 respectively.
Chris