Look for a bag in the range of 14 to 15 inches wide by 12 inches high.
Choose one with plenty of pockets - inside and out - including an outside pocket for airline tickets. Most of the outside pockets should be zippered to keep items from falling out.
Choose a roomy bag that will hold as much as possible, but still keep things accessible without a lot of digging.
Find a carry-on bag that will ride "piggyback" on a wheeled suitcase by attaching to the handle. This makes it easier to transport and keeps hands free.
Make sure the strap is wide and fits comfortably over the shoulder. The strap should also be adjustable.
Choose a color that goes with the rest of the traveler's luggage.
Make sure the bag has a luggage tag. If not, buy one and attach it.
Consider having the bag monogrammed as an nice touch.