I understand, to some extent at least, why lions are an important symbol in Judaism. After all, in the beginning of the Shulchan Aruch we are told we should rise like a lion to serve our Creator, and lions are often mentioned positively in Tanakh. Aryeh (and Leib) have long been common Jewish names. But why Dov and Dov Ber? What is the symbolism or significance of bears in Judaism?
Rav Mirsky in his first volume of Hegyonei Halacha has an interesting article on Ameilah shel Torah and includes the virtues of a bear. In speaking about how important 'toil' in learning is (rather than rote learning) he brings a Radak on Hosea (13:8):
.אֶפְגְּשֵׁם כְּדֹב שַׁכּוּל, וְאֶקְרַע סְגוֹר לִבָּם; וְאֹכְלֵם שָׁם כְּלָבִיא, חַיַּת הַשָּׂדֶה תְּבַקְּעֵם
I will meet them as a bear that is bereaved of her whelps, and will rend the enclosure of their heart; and there will I devour them like a lioness; the wild beast shall tear them.
The Radak asks why Hashem's anger is compared to a bear rather than a lion, the 'king of beasts'. He says that a bear's offspring is born with a very thick amniotic sack and it takes much care, toil and difficulty to bring it in to this world; more so than any other animal. So much so, that when a bear loses a cub it experiences greater anguish and loss. He finishes off by linking this to Torah in a parallel way: that only things that require hard/devoted labour reaps reward.
This doesn't link naming a person Dov/Ber but is a relevant insight into the virtue of a bear, at least according to the Radak. I assume people naming children after animals are doing so according to traits and essences of that animals.
According to Alexander Beider's Handbook of Ashkenazic Given Names, Dov didn't become a name in "the vernacular life" until the 20th century. "Jews called Dov in Hebrew sources were actually named Ber in their everyday life."
Ber, on the other hand, comes form the German Bero which has been known since the 8th century among non-Jews. Beider's theory is that it is a hypocorism for Bernhard.
German Jews borrowed both the non-compound form Ber(o) and the version Berman (bear + man). This borrowing is likely due to (1) the attractive meaning of this name (bear implies force, endurance, virility); (2) the popular use of these names by Gentiles.... [The great popularity of the name Ber] is explained by its attractive meaning... cf. Leyb, Hirsh, and Volf, other male names connected to animals. The popularity of Ber also contributed to the creation of the new Hebrew appellation, Dov, the Hebrew calque of Ber.