Renting car without owning credit card

Contact your credit card issuer, and ask for a second card on your account with your wife's name printed on it. Most credit card companies will happily do this.

Contact a local car rental company instead of international companies. Talk to them and ask how they can handle such a case. I am pretty sure local companies in Iceland are more able and willing to handling such a case.

With the big car rental companies, local agents often have restricted responsibilities and possibilities for handling more complex cases. I made my experiences...

Or book your car via a travel agency, then you may not need a credit card at all.

Here are a few alternative options to the other answers:

1. Buy a pre-paid credit card as per Count Iblis's comment, but ensure that there are sufficient funds for them to pre-authorise the deposit. Call the car rental company in advance to check that (a) they will accept pre-paid cards, and (b) to find out how much they will attempt to pre-authorise so that you can ensure you have sufficient funds on there. Be aware that if you use this option, you will need to find a subsequent use for the excess funds that you had to leave on it to cover the pre-authorisation. Most cards will let you withdraw funds from a pre-paid card, although many charge a fee for this.

2. Pay using your credit card, and use cash or your wife's debit card to make the deposit when you arrive. Call the car rental company in advance to make sure they will accept this. I have personally done this in the UK. I paid using a friend's credit card, then presented my own debit card upon arrival. They charged me £500 which they returned when I dropped the car off (because pre-authorisations do not work with debit cards). This works because paying for the car is a separate transaction to presenting your credit card upon arrival. Or at least it is if you use Expedia (and presumably, other similar sites):

The driver must present a valid driver's license and credit card in their name upon pick-up. The credit card is required as a deposit when renting any vehicle. The deposit amount is held by the car hire company. Please ensure sufficient funds are available on the card.

3. Have your wife apply for a credit card. Unless she has a bad credit rating, this shouldn't be too difficult. If she has a bad credit rating, then select a company that specialises in that. They usually have very high interest rates, but that isn't a problem as long as you only use it for its intended purpose i.e. book the car, and repay the balance immediately. If you need to make the car rental booking in a hurry, then pay with your card as per option 2, then apply for your wife's card and bring that with you for the deposit.