If $ n $ days have passed since Sunday, then the total number of cents in her bank account is $ 1+2+\cdots+2^n $. This is a geometric series with first term 1, common ratio 2 and $ n+1 $ terms. Hence the sum is: $$1+2+\cdots+2^n = \frac{1-2^{n+1}}{1-2} = 2^{n+1}-1.$$If this is greater than $ 500 $ (i.e.$\phantom{}$ if the total amount of money in the account is more than $ \$5$ ) then $2^{n+1}-1\ge 500$ , so $2^{n+1}\ge 501$ . The smallest power of 2 that is greater than 501 is $2^9$ . Thus the first time there is more than $\$ 5 $ in the bank account occurs after $ n=8 $ days. This is 8 days away from Sunday, so the day of the week is $ \boxed{\text{Monday}} $.