First, welcome (almost) aboard.
I am assuming you personally made payment for your account. If you paid with PayPal, it may take more time (search in this forum for PayPal).
If you paid by check, it may take even more time.
I think the “one business day” means after DH (DreamHost) knows they actually have payment in possession, they will activate your account “very soon”.
On the other hand, if your friend has an account here and is also paying for your account, you (both) might run into an issue. DH provides a lot of space and bandwidth for a very small amount of money, but does charge for going over those generous limits. If it looks to DH like your friend is trying to get a second account, then you should both read the terms of service very carefully.