The whole idea of a "cost basis" is basically that it's money that you've already paid taxes for. For instance suppose you work at a job and get $1000, and you use that money to buy stocks, and several years later the stocks are worth $1500. You already paid income tax when you made the $1000 at your job. So when you sell the stock, you only have to pay taxes on the extra $500.
In this case, you haven't paid taxes on any of the money, so there's no apparent reason for there to be a cost basis. There could be one, however, if there are other facts. For instance:
If you claim that the computer was a gift, then the client would have to deal with the tax implications of that, while you would get the FMV at the time of gifting as cost basis. It would be difficult to claim that your client giving you stuff isn't in consideration for your services, however, and it's unlikely that they would want to deal with treating it as a gift.
Also, if you declare the computer as income in one tax year, and sell it in another, the FMV when you got it would be taxable income, and then you could claim that as a cost basis in a later year when you sell it.
Then there are further situations, such as depreciation, etc. However, without any further information, there's no reason to think that cost basis is relevant.