Learn Russian: Я хочу дать тебе один полезный деловой контакт. - I want to give you a certain useful business contact.

Not exactly. Here, один is used as an indefinite specific pronoun, meaning 'a certain' or 'a particular'. The speaker has a specific contact in mind but is introducing it as new information to the listener. It is much like saying 'I have a certain business contact for you.'

Since a 'business contact' refers to a person, why doesn't it take the animate accusative endings (одного полезного делового контакта)?

The noun контакт is grammatically inanimate, regardless of whether you are talking about an electrical contact or a professional connection. Because it is a masculine inanimate noun, its accusative form (the direct object of дать) is identical to its nominative form.

If the speaker is introducing a contact unknown to the listener, could they use какой-то instead of один?

They could, but it would change the meaning. Какой-то usually implies the speaker does not know the person either ('some unknown contact'). By using один, the speaker indicates that they know exactly who this contact is, even though the identity is still new to the listener.

Can I say бизнес-контакт instead of деловой контакт?