The Star Wars in 100 Scenes factbook confirms that his request to shut down was guided by it aiding his mechanical systems in repairing his damaged arm.

The official Alan Dean Foster George Lucas novelisation of 'Star Wars: A New Hope' backs this up and gives us some additional dialogue that didn't appear in the film.

“Sir?” came a familiar voice Luke hadn’t heard in a while. “What?” Luke was startled out of his examination. “If you’ll not be needing me,” Threepio declared, “I think I’ll shut down for a bit. It will help the armature nerves to knit, and I’m due for some internal self-cleansing anyhow.”

The Star Wars NPR Radio Dramatisation (now non-canon but still good fun to listen to) contains a similar line: