How do I legally change my name to a Muslim name?

First: you don't have to

Most of the Companions kept the names their parents gave them. A name only needs to change if its meaning is bad or it suggests worship of someone other than God, and almost no Western name falls into that group. Many converts keep their legal name and simply use a Muslim name among friends at the mosque. See Do I have to change my name after becoming Muslim?

How it works in the UK, US and Europe

• UK: adults in England and Wales usually make a deed poll, a signed statement witnessed by someone who isn't family. You can write one yourself for free or pay a deed poll service. An "enrolled" deed poll through the courts is optional and only needed for some purposes. Scotland and Northern Ireland have their own routes.
• USA: you file a name change petition with the court in your county. Rules, fees and whether you must publish a notice in a newspaper vary by state, so check your state court's website.
• Europe: many countries handle it through the civil registry or a court, and some ask for a reason. Converting to Islam is often accepted as a reason, but rules differ widely.

Your mosque's shahada certificate can help as supporting evidence, though it is not a legal document. See Do I have to register my conversion to Islam?

After the legal change

Update your passport, driving licence, bank, employer, doctor, tax office and any professional bodies. Keep certified copies of your deed poll or court order, because you'll be asked for them often. Old degree certificates usually stay in your old name, and that's fine.

What about your surname?

Changing a first name is simple. Scholars are more careful about surnames, because Islam values knowing who your father is:

So avoid a legal name that falsely presents someone else as your father, such as "ibn Abdullah" when your father is Mark. A common choice is to add a Muslim first or middle name and keep the family surname, which also keeps the link with your parents.