The name Deborah in the Bible
There are two Deborah's mentioned in the Bible. The first one is the nurse of Rebekah (Genesis 35:8). The second one is a judge in Israel and war hero, famous for her role in the war against Jabin, king of Canaan (Judges 4:4).
We don't know much about the first Deborah, except that she probably breast-fed Rebekah when she was an infant—the word for nurse comes from the verb ינק (yanaq), meaning to suckle—and went with her when she left home to be with her husband Isaac (Genesis 24:59, 61).
Deborah number two had her own palm tree—fittingly called the palm tree of Deborah (Judges 4:5, but also see Genesis 35:8)—was a prophetess, and was probably married to a man named Lappidoth, although since the Hebrew text of verse 4 only states that she was a woman from or of Lappidoth, it might also have been that she simply came from a town named Lappidoth (says BDB Theological Dictionary). No town called Lappidoth is ever mentioned in the Bible, but then, no husband named Lappidoth either.
Etymology of the name Deborah
The name Deborah is identical to the word דבורה, meaning bee, and both come from the magnificent root דבר, meaning to speak or pronounce:
Tradition and most Bible commentators and translators (including Jones' Dictionary of Old Testament Proper Names) assume that the bee was known after a derivation of the verb דבר, meaning to speak, because of the sound a bee makes when it flies. This is highly unlikely for two reasons.
First, the bee is not the only creature that makes a sound, or even buzzes. Calling a bee but no other insect a talker would show a imprecision that is ultimately foreign to the Hebrew language.
Secondly, even in Bible times, the bee was culturally defined as a producer of honey. Honey was the only available sweetener in those days, and honey was recognized as a great source of strength (1 Samuel 14:27). Where milk is compared to the initial nutrition of a new believer (1 Peter 2:2), honey serves the die-hards. Hence Canaan was known as the land flowing with milk and honey (Exodus 3:8), and the judgments of the Lord, as well as His words, were deemed sweeter than honey (Psalm 19:10, Psalm 119:103). King David reckoned unity in the House of the Lord sweet (Psalm 55:14); Ezekiel tastes a scroll that was given to him by The Word Of God, and it tastes sweet as honey (3:3), and the same happens to John the Revelator (Revelation 10:10).
Another prominent insect in the Bible is the fly, which also buzzes. The Hebrew word for fly is זבב (zebub), which serves as segment of the name Beelzebub, meaning Lord Of The Flies. The difference between the kingdom of God and the whatever-dom of satan shows clearly in the difference between the bee and the fly: bees have a house, live in a colony and adhere to central rule. Bees like flowers, make honey, speak a language, care for offspring, and are armed. Flies are homeless, aren't social, acknowledge no authority other than themselves, like dung and decaying flesh, make nothing, speak no language, don't care for their offspring, and are not armed.
So yes, the name Deborah translates as Bee but it's a name with a very large and indispensable footnote.