What is the value of a 2003 US dollar bill with an A mint mark?

To clear things up, only coins have mint marks. US paper money is printed by the Bureau of Engraving and Printing rather than the Mint.

If the "A" is in a large circle on the left side of the bill, it's a code letter for which Federal Reserve District distributed the bill; "A" refers to the Boston District. A 2003 $1 bill wouldn't have a series letter, but other dates can have a small letter next to or below the series date. In that position, an "A" indicates that the bill was part of the second printing run under the administration of the same Treasury Secretary but the second US Treasurer.

Note that bills aren't printed by each Federal Reserve district, just distributed through them. There are only two printing facilities, one in Washington and the other in Fort Worth. Bills printed at the latter facility have a tiny "FW" next to one of the plate-position numbers on the obverse. In general there is little difference in value between bills issued by the two locations.