I heard this term なびっくる live on http://www.fm-totsuka.com/ I checked http://dictionary.goo.ne.jp/ and http://ejje.weblio.jp/ but could not find any entry for なびっくる, https://translate.google.com/#ja/en/%E3%81%AA%E3%81%B3%E3%81%A3%E3%81%8F%E3%82%8B has it as "surprised". Is なびっくる a variant of びっくり?
No なびっくる is not a variant of びっくり. Perhaps you've misheard something. Since you've heard it on the radio, is there a possibility that you've heard an ad of ナビクル (There's an ad library)?