What is the Biblical basis for saying Lucifer is an angel? Where in the Bible does it say that Lucifer was one of the angels or an archangel?
Interesting question. In the Bible, this is strongly hinted at rather than directly stated. Here is the evidence:

In Job 1:6 we learn that Satan was in heaven with other angels.


  One day the angels came to present themselves before the LORD, and Satan also came with them.


That Satan is (was) in heaven is also seen in Zechariah 3:1-2, Luke 10:18 and Revelation 12:9. Revelation 12:9 gives a further clue to Satan's identity:


  The great dragon was hurled down—that ancient serpent called the devil, or Satan, who leads the whole world astray. He was hurled to the earth, and his angels with him.


This thought is also stated by Jesus in Matthew 25:41:


  “Then he will say to those on his left, ‘Depart from me, you who are cursed, into the eternal fire prepared for the devil and his angels.


That he has his own angels, is a further clue to him being a (former) angel. (And incidentally, Isaiah 14:12-15 is often used to show why Satan fell, because he wanted to ascend to God's throne).

Note also 2 Corinthians 11:14, which says that Satan can "masquerade as an angel of light."

Conclusion: we know that Satan lives / has lived in heaven, with other angels (some of which are his own), that he was (or will be) cast out of heaven. From this we infer that he is a "fallen angel".

Update: In my answer above I assembled evidence for Satan being an angel. But it's not proof (see comments below). Which leads me to consider proof by exhaustion: if Satan was not an angel, what could he be? I know of only three types of sentient beings in the Bible: God, man and angels. He's clearly not God, nor man. That only leaves one other option.

Update on Lucifer: @Dan asks where Lucifer is named as such. The answer is in Isaiah 14:12 KJV:


  How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations!


This verse and the wider passage are historically interpreted as referring to Satan, though it is not directly stated.

Footnote: there are only two named angels in the Bible. Michael (Daniel 10:13, Daniel 10:21, Daniel 12:1, Jude 1:9 &amp; Revelation 12:7) &amp; Gabriel (Daniel 8:16, Daniel 9:21, Luke 1:19 &amp; Luke 1:26).