Firstly, Newt Gingrich is defeated for re-election in 1990. Then, a weaker Democratic nominee is put up in 1992 and Bush is able to win re-election. The Democrats, naturally, keep the House in 1994, and in 1996 someone like Cuomo or Chuck Robb cruises to victory over a hapless GOP ticket headed by Dan Quayle.
Their first term is relatively uncontroversial. The GOP boots out Bob Michel and replaces him with Dick Cheney, who isn't a fantastic leader, especially when having to deal with sex scandals from both Bob Livingston and Dennis Hastert. There's a second Gulf War in 1998 following a terrorist attack, allowing Democrats to mitigate their losses in the midterms. Dot-com bubble bursts and though the Democrat wins re-election, it's by a closer margin than expected and control of the House is cast into doubt. Withdrawal from Iraq occurs in 2001 at the same time as rapprochement with Iran, but even good news can't spare the Democrats from losing control of congress for the first time since Truman.
The Democrat spends the rest of their term passing compromised legislation, including a tight ban on corporate contributions. The Republicans win back the White House in 2004 under a hawk who wants war with Libya, and gets it. The war quickly goes south and there is little international support for the measure. In addition, a rapid series of tax cuts and deregulation passed while the party still holds both houses ignites controversy and inspires liberal activists, who are beginning to use "netroots" to their advantage. 2006 sees a backlash against this president, and Congress returns to a more natural shade of blue. 2008 sees them booted out after a housing crunch.
I'm a bit bored now so have a list idk
1989-1997: George H. W. Bush (Republican)
1997-2005: Chuck Robb (Democratic)
2005-2009: Norm Coleman (Republican)
2009-2013: Martin Chávez (Democratic)
2013-2017: Asa Hutchinson (Republican)
2017-0000: Susan Bysiewicz (Democratic)
1989-1999: Tom Foley (Democratic)
1999-2003: Martin Frost (Democratic)
2003-2006: Tom DeLay (Republican)
2006-2007: Don Nickles (Republican)
2007-2013: Diana DeGette (Democratic)
2013-2015: Roy Blunt (Republican)
2015-: Diana DeGette (Democratic)