I am putting the finishing touches on a novel taking place in 1998, and want to mention singing a specific children’s song rather. It has to be a relatively new song in that time period, but I would prefer if it wasn’t Barney 
(oh, and should also be appropriate for ages 4-whatever time non-jaded children stop enjoying kids songs.)
Didn’t Pinky and Brainbegin in 1998?
Blue’s Clues theme song? Started in '96, but was still popular then.
Might also want to check out this site: http://www.kidzmusic.com/nappa1998.htm
A savvy 4 year old might have been into Laurie Berkner by 1997 - most of the rest of us found out about her by 1998. “We Are the Dinosaurs” is painfully catchy.
Raffiwas (and is) still popular with the preschool set. “Baby Beluga”, “Bananaphone” and “Down By the Bay” probably being the most famous at our house.
Here’s a list of the 1997 NAPPA Awards For Children’s Music. Anything you’ve heard of should qualify as “famous”.
“The Wheels on the Bus Go Round and Round” was very popular when my now 11-year-old son was a wee babe.
Not a kids’ song per se, but my little cousin couldn’t stop singing “Barbie Girl” by Aqua that year. Also “Tubthumping” by Chumbawumba, not appropriate, though. “She drinks a whiskey drink, she drinks a vodka drink…”
I remember reading an article about the Macarena dance craze. The author said if you have no idea what it is, just ask a nearby 5-year-old. The Macarena was a few years earlier than '98, however.
1995, actually, although the characters themselves first appeared on Animaniacs, which debuted in 1993.
Besides Blue’s Clues, if you want to go with a kids show, you could go with Teletubbies- they debuted in April of 1998 in the US to much hoopla.