Something I have yet to see (or hear, I guess) from a mixer is a song that I truly liked before it got remixed. Some of them have been songs I liked, or was okay with, but most are songs that I didn't like to begin with. (It might be because I prefer various types of rock, which probably wouldn't make great mixes).
My recommendations:
What We Came Here For and Out Here All Night, by Damone
Not the Girl, by Rachel Farris
Anything from Within Temptation (I love All I Need the most, but it'd probably suck as a remix; I could be wrong)
Do You Want A Taco?, by Psychostick
Nothing to Do With Love and Bet You Wish You Had Me Back, by Halestorm
Scream, by Michael Jackson
Where in The World is Carmen Sandiego?, by Acapella (That could be awesome!)
Anything but Ordinary, by Avril Lavigne
Merry Go Round, by Puddle of Mudd
My Japanese recommendation:
Kagirinai Yokubou No Naka Ni, by Megumi Hayashibara
Mo Nakanaide, by Azusa Senou (I've seen the "Mo" as "Mou" as well, so you might wanna look out for either option)
Kimi Ni Mune Kyun; not sure who officially did this song, but it's the ending theme from Maria Holic
Lost My Music, by Aya Hirano
Little Date, by Ribbon
Clover, by Meg Rock
Yasashii, Ii Ko ni Narenai, performed by Hidaka Noriko (This song is better known in the Sates as The Baka Song)