Working In a Goldmine (Aztec Camera)
Rain (Beatles)
Senses Working Overtime (XTC)
This Is A Low (Blur)
Big Me (Foo Fighters)
Down Down (Status Quo)
Sunshine On Leith (Proclaimers)
Born Slippy (Underworld)

I saw Glen Tilbrook play a solo acoustic set, just him and a guitar, in a small venue a few years ago. His own stuff was, of course, top notch but the highlight was when he launched into this song, which I think satisfies your criteria. Both familiar, yet revealed in a new light when not performed by its original composer. Once the associated baggage/prejudice/knickerbockers and cap etc. are stripped away its true beauty and magnificence shines through. Fairly basic chords too, I think.
Brought the house down when Glen did it. So here's hoping it does so for you.

always goes down well, from experience of being in an acoustic duo that plays in pubs. Also, surprisingly, Shoot Out The Lights, Fever, These Boots Are Made For Walking, Two of Us and The Bluetones' Slight Return. And Into The Mystic.

by Blur?
I'd also go for Day We Caught The Train by Ocean Colour Scene, but I fear I may be alone on that one.

Dexy's Midnight Runners.

....stop banging on about Gilbert O'feckingSullivan, I will desert for Unshod and Slomo forever.
(Is that someone opening the door? How thoughtful.)
Be worried: Tom Jones seems to have been rehabilitated, Gilbert seems next. Bear in mind that the 3rd stable member of Gordon Mills management was Englebert Humperdinck....... I await the first unbridled praise for Last Dance, as in Save the.

There's a local band called The Squirts from Swansea who's setlist is always interesting. The play all covers but often the less expected songs of major artists. They play a lot of Dylan but would play "Positively 4th Street", "When I Paint My Masterpiece", "Desolation Row" etc. They do a storming version of "Dead Flowers" by the Stones as well. They've turned me onto a lot of artists by playing their songs like Richard Thompson (Wall of Death) and Nanci Griffith (Gulf Coast Highway) for example.

If you're ever in Swansea and they're playing their well worth a watch,and no I don't play in the band I just like them.

than check out Thea Gilmore's covers of Ever Fallen In Love and You Spin Me Round.

Here's your surefire showstopper. They'll be hailing you to the rafters. All-time pop classic. Lends itself to acoustic rendering and sounds brilliant in a different way in that context. You'll be knee-deep in lasses, son.