I just did the opposite on my son's bike...swapped the twist shift for a trigger shift, paired with the stock 6-speed Tourney RD. I found a company called S-Ride that makes Shimano-compatible drivetrain components. They have a 6-speed RD that might be a decent (and cheap!) replacement:
www.universalcycles.com
The quality of the trigger shifter is pretty good...and is a much better ergonomic match for my son.

S-Ride M200 Rear Derailleur - 6/7/8 Speed; Shimano 2:1 Ratio
S-Ride M200 Rear Derailleur - 6/7/8 Speed; Shimano 2:1 Ratio :: From $11.00 :: S-Ride Rear Derailleurs

The quality of the trigger shifter is pretty good...and is a much better ergonomic match for my son.