I came across a website which claims
beren means "bear" in early Low German. That would make Berengar a dithematic name meaning "bear spear".
Unfortunately the website is now defunct, so the link below points to Google's cache:
http://www.google.com/search?q=cache:BIpcy4-chSU:www1.tip.nl/~t401243/names/name4_en.htm+beren+germanic+root&hl=en