Best place to buy souveniers is not at Disney. We buy our souv's before we go.
Walmart sells a lot of fun little item with Disney characters, pens with bobbing Disney character and fuzzy stuff on end were $1, party aisle has great Disney character items that are good for kids for a couple bucks a package (make neat gifts for kids to give to their friends after they get back). Also, Target carries Disney character t-shirts, pajamas, socks, etc...
Check out the clearance section at your local
Disney Store if you have one also. If you are just looking for t-shirts consider going to a Florida Walmart or Walgreens and get "Florida" t-shirts that are $3-$4 each. Most of our friends & kids have been just as happy with Florida shirts as with Disney shirts.
Also, go to Walgreens and check out their selection of shell/beach inspired jewelry (think puka shell necklaces of 1970's). They are almost identical to the ones sold at Disney gift shops and cost about half the price.
Downtown Disney has an Under $10 gift shop. We buy a couple of grab bags for about $6-$7 each year. They contain a lot of cute little items and are worth the money. This year our bags had a magnetic postcard, Tinkerbell note pad & clip board, wind up Donald Duck, Timon & Pumba matchbox type car, etc...