I found this forum discussion from a few years ago, which stated:
LH Express was their brand name for domestic services for some years in the mid 90s. I'm not sure when the LH Express name disappeared, but they definitely were not promoting the Express name in 1997 any more. So the aircraft you saw was probably just not repainted yet. LH used to operate B737 of various series and several Airbuses.
One of the guys went on to say it was a low-cost domestic route, set up in competition to Deutsche BA. Unfortunately, it did not succeed enough in competition and was eventually shut down. They did fly some small Boeings like the 737, and the Airbus 300 and 310.
Additionally, a NY Times article from 1992 gives a small blurb about the startup of the line:
Under the new strategy, eight key domestic German routes are to be served by a special spin-off division, Lufthansa Express, that promises passengers a better on-time record, faster check-in and a chance at bargain prices.
Another innovation of Lufthansa Express will be bargain prices on late-night and short-notice flights. Passengers get a 55 percent discount savings on flights after 7:30 P.M., even without advance reservation, for example.
The most controversial aspect of Lufthansa Express is something passengers are unlikely to hear much about: The starting pay for a crew on Express flights will be about 25 percent less than on traditional Lufthansa flights.
Another thread here has a few minor other details. Interestingly enough, Amazon still has a listing for a model LH Express 737 (out of stock).