Bask Bank
Bask has a referral program where the new account holder and the referrer each get 5,000 miles. The referrer can earn 25,000 maximum via referrals. Existing account holders can provide a referral link.

Archived Bank Direct offer below, no longer valid:
Bank Direct offers American Airline miles for an online bank account. Miles are available for an online Checking account ($12 per month) or a Money Market account (no monthly fee).

The "Refer A Friend" program allows existing AA earning account holders to refer somebody. Both the existing AND the new customer get an extra 1000 miles for the referral. The existing account holder must log in, choose "manage my account", then "refer a friend".

All that is needed for this referral is the new customers email address. Addresses may only be referred once, so if you sent an email to someone ages ago and they never opened an account, you cannot send another referral to that friend's same email address.

Here is the referral screen that the existing account holder will see: Notes on the checking account bonus:

• Earn 1,000 miles for being referred
• Earn 10,000 miles for direct deposit
• Earn 5,000 miles for using bill pay
• Earn 5,000 miles for using your BankDirect Visa CheckCard
• Earn 100 miles per month for every $1k average daily balance up to $50k (above $50k earn 25 per $1k).

That's at least 21,000 miles!

• Another nice feature is you get 4 reimbursed ATM fees up to $2.50 each.
• Miles post around the 3rd or 4th of each month. Bonus miles post the month after you meet the terms (for me at least).
• Note that the direct deposit must be $2000 per month for 6 months to earn the bonus.
• If you want to keep your account active without thinking about it, set up a recurring payment of $1 to some credit card.