The phrase "right lane" is correct and usable in written English.
You can use it when referring to a lane on the right side of a road or highway. For example: "I need to merge into the right lane to exit the highway."
Seconds later, Ghaisar's Jeep stopped in the right lane.
For reference, purified adiponectin is shown in the right lane.
The car was in the right lane and the patient was okay, so we only closed the right lane.
Then on the right lane of the northbound Major Deegan Expressway, he pointed out the deep trenches running along the right lane.
From the Mass Pike (I-90), take exit 18 (Allston) and move into the right lane.
Take exit 26 for Storrow Drive, using right lane and following signs for Storrow Drive.
Proceed through the underpass and move into the far right lane which feeds into Massachusetts Avenue.
A few lights later, a taxi broke down in the right lane, further complicating the mess.
In the far right lane, a semi is coming up to pass me.
"Stephen stayed in the right lane the whole way," Jones said.
