Most legal documents are named for what they are. The affidavit is named for something a person did. It is a Latin verb in the past tense, frozen and turned into a noun.
In Medieval Latin, affidare meant "to trust" or "to make an oath." It joins ad, "to," with fidare, "to trust," which comes from fidus, "faithful." The ad turns into af before an f, which explains the double letter. Affidavit is the third-person singular perfect form: "he has stated on oath."
Etymonline says the document is said to have taken its name from being the first word of such sworn statements. That is a reasonable account of how a verb ended up as a label. The phrasing "said to be" is worth keeping, since no source quoted here shows an early document opening that way.
English had the word by the 1590s, in the sense "written declaration upon an oath," according to Etymonline. Dictionary.com dates its first record a little later, to 1615-25. Either way, it entered English as a lawyer's term and has stayed one.
The modern affidavit still works as the Latin describes. Someone makes a written statement under oath or affirmation, and it serves as evidence of what it contains. The person swearing is called the affiant or deponent. A notary public or commissioner of oaths witnesses the signature, and lying in one counts as perjury.
Some systems now skip the oath itself. Wikipedia notes that US federal courts, and about 20 states as of 2006, accept unsworn declarations made under penalty of perjury instead.
The trust at the heart of affidare runs through a large English family. Latin fides, "trust, faith," comes from the same ancient root, and so do fidelity, confide and bona fide. The closest relative is affiance, borrowed in the 1520s from Old French afiancier, "to pledge, give one's word." Its Old French source traces back to the same Late Latin affidare. A sworn statement and a promise of marriage are, at the root, the same act of pledging faith.