The Republic of the Rio Grande Museum is definitely worth the visit (if you like history and stuff. If not, just go to the mall, or the produce section at HEB). It's a very small museum, takes 30-60 minutes to see, but it's crammed full of history from the early 1800s about Laredo and the Republic. It's why there have been 7 flags to fly over Laredo instead of the standard 6 that have flown over Texas.
The country existed for 283 days in 1840 when insurgents rebelled against Santa Anna and the Mexican government. Parts of 3 northern Mexican states and the areas around Laredo and Victoria made up the new country, with Laredo being it's Capitol.
I think it's one of the most interesting ways you can spend four bucks and kill an hour or two in Laredo. And it's in a nice area of town.