Unfortunately they cannot pay more than amount on the invoice but, they can pay a partial amount - If you want to accept a deposit or partial payment from your customer, go to the Create Invoice page and check the "Allow partial payment" box at the bottom left of the page. You can also set a minimum amount (optional). Your customer can then make a full payment, pay the minimum or pay an amount greater than the minimum. Once paid, your invoice will have a partially paid status until it’s paid in full. The amount paid will also appear on the invoice for both you and your customer.
A possible work around would be for the customer to send you the difference via the Send Money feature and select "goods and services" - they could pay the invoice first and then use the send money feature to pay the balance.