A birthday is just as special for a Muslim boy or girl as it is for anyone else. There are no rules in Islam for conducting birthday parties, so Muslims usually celebrate them in accordance with their local customs. A birthday party for an American Muslim would be no different, but some modifications are possible in accordance with fatwas that have been issued to ensure that a celebration of one's birthday doesn't leave God out.
Begin with a short and informal duaa (prayer). Anytime that Muslims are gathered together it's customary when addressing a group of people to begin by saying "bismi Allah al-rahman al-rahim" and extemporizing your gratefulness and thanks for blessings. Since it is universally agreed that celebrating a birthday as an act of worship is prohibited, there's no need for a formal or Arabic duaa such as one recites for a sermon or on a religious festival (“eid”) day.
Encourage guests to make a small donation to a charity of your choice, either in place of gifts or -- if you want the birthday to be any fun at all -- in addition to them. It is common in Islam to give something to charity to express appreciation to God. Make a donation yourself, as well.
Give your guests something nice. Hospitality is an important part of Islamic adab, or etiquette. Again, there are no formalized rules here, but it's very much a Western convention for your child's friends to brings presents. There is nothing prohibited in this, but a little bit of reciprocity can only liven things up.
Throw the birthday party as you would otherwise. Get delicious cake, put up streamers, and make sure there are lots of games to play.
Some Muslims consider the celebration of birthdays in general to be prohibited, on the grounds that there are ought to be no other eids (feasts) than religion dictates. Others assert that a birthday is not an eid, but merely a thankful celebration. If you feel uncomfortable celebrating a particular day, you can also move the celebration by a day or two, as often happens anyway, to have a birthday on a weekend day.