Yes, it is okay for Christians to sing "happy birthday." The song itself contains nothing but the words "happy birthday to you" and the name of the person whose birthday it is, so the issue really comes down to whether birthdays are acceptable at all. If birthdays are okay, then it would obviously be okay to wish someone a happy one.
Are Birthdays Morally Permissible?
Jehovah's Witnesses have long asserted that birthdays are actually forbidden in Scripture. They base this claim on the fact that the only birthdays mentioned in the Bible are those of Pharaoh (Genesis 40:20-22) and the wicked King Herod (Mark 6 21-28, Matthew 14:6-11). The Jewish historian, Josephus, also records of the Roman Emperor Titus honoring his brother's birthday by slaughtering Jews.1 They note that not only are these celebrations all by pagan kings and not God-fearing Jews or Christians, but also that they all involve parties that devolve into people being killed. In the case of Herod, you also have the scandal of Herod's lusting after his wife's daughter and the fact that the one who was killed was a prophet of God.
It is true that the only birthdays mentioned in Scripture are those of pagan kings, but that does not automatically condemn all birthday celebrations. The only coppersmith mentioned in Scripture is a man named Alexander who brought great harm to Paul and upon whom Paul pronounces God's judgment (2 Timothy 4:14), but that does not mean that it is inherently evil to be a coppersmith. The Bible is not condemning all who craft copper pots and tools as wicked pagans simply because it never mentions a righteous one. Alexander just happened to be a coppersmith, so that is how Paul distinguished him from other men named Alexander. In the same manner, birthday feasts in honor of pagan kings were the setting of these particular scenes, but that is not the focus. The Bible makes no comment on the practice of celebrating birthdays itself. One is not meant to read the account of Herod and walk away focused on the fact that he celebrated his birthday. The focus is on the lust, treachery, deceit, and murder. These are the sins condemned in this story. A birthday feast is merely where this happened to take place.
It is, of course, possible to celebrate birthdays sinfully. If one is using one's own birthday as an opportunity to honor and aggrandize oneself, or to greedily and selfishly covet expensive gifts, or (as in the case of Herod or Titus) to engage in all forms of debauchery in the name of "partying," then one is celebrating a birthday in sin. The sin, however, is not the birthday itself. The birthday is merely the occasion. The sin is the pride, covetousness, sexual immorality, drunkenness, violence, or the like. Likewise, some early Christians expressed concern about paying attention to birthdays because of the connection to astrology and fortunetelling2 (as with today's horoscopes). Again, however, the sin here is occult practice, not birthdays in and of themselves. If one finds birthdays to be a temptation for any of these things, it is probably best to drop the practice.
"If your right hand makes you stumble, cut it off and throw it from you; for it is better for you to lose one of the parts of your body, than for your whole body to go into hell" (Matthew 5:30).
Likewise, if one has a burden in one's own conscience that birthdays are not something Christians should celebrate or pay attention to, it is best not to participate. If you believe something to be wrong and do it anyway, then you are doing it with a heart of rebellion and thus you are sinning even if the thing itself is actually permissible.
If, however, you regard birthdays as an opportunity to give thanks to God for another year of life or for the life of a loved one, then there is no harm in rejoicing in humble thanksgiving to God for His blessings. There is nothing wrong with commemorating each year the anniversary of some great gift of God in one's life. Even our Jehovah's Witness accusers celebrate wedding anniversaries. Celebrating someone's birth is no different. If we do it as an act of thanksgiving, giving glory to God, then it is good.
"If I partake with thankfulness, why am I slandered concerning that for which I give thanks? Whether, then, you eat or drink or whatever you do, do all to the glory of God" (1 Corinthians 10:30-31).
Since celebrating someone's birth violates no explicit command nor any inherent moral principle of Scripture, we are free to do so if we do it without sin and with humble thanks to God.
While birthdays are not established in Scripture, they are also not forbidden. There is nothing inherently wrong with celebrating the anniversary of a loved one's birth through song, gift-giving, food, and fellowship. Indeed, if it is done with a humble heart of thanksgiving toward God for the life He has given, it can be a healthy expression of Christian liberty for those who choose to do so. And if it is okay to celebrate a birthday in all these ways, it is certainly okay to sing "happy birthday" to someone.