I think it's fairly common to have it these days. The father/daughter and mother/son are certainly going to be one of the more awkward moments of the wedding. I would suggest the following:
- A) Tell the DJ to cut the song short, 3-5 minutes can feel like a lifetime, 1.5 to 2 minutes is plenty of time to get pictures in and move on. Most DJ's will generally ask you if you want to do a short version
- B) Do both the father/daughter dance and mother/son dance at the same time. Two birds with one stone, minimizes awkwardness.
Song Choice:
Beach Boys - God Only Knows (There are a ton of covers as well, Natalie Maines of the Dixie Chicks does a pretty solid one)
Spice Girls - Mama (real on the nose and super cheesy)
Sidenote, you can literally skip any of these "traditional" parts of a wedding almost no one will notice or care. The only thing people will remember is if the DJ was bad (just tell them to play a lot of 90's pop and classic rock and asses will be on the dance floor all night) or if there was no booze.