Pet Fish

Just as a suggestion, think of how big of a tank you want, and then think "layers" - get a mix of some bottom type fish, then middle layer lovers and top swimmers.  It makes the tank more lively.

Things to make sure to have - tank, aeration system, bottom covering (like colorful, small rock (helps with aeration), decor (like fake plants or coral, etc.), heater, temp gauge, food,
unless you just want the small tank with clear marbles and a goldfish.

Since it will be your tank that she will be looking at, anything really.

Pretty much all common freshwater fish are pretty hardy.... that is why they are common. Gold Fish and Plecos can survive pretty much anything. Get a Gold Fish, fill up a bowl, and change the water every so often if that is all you want. After that, well, you sort of need to learn some things and how to keep healthy fresh water fish with all the other stuff mentioned above. You can have a really nice planted 30g fresh water tank with nice healthy fish without it costing a fortune... a reef tank.... well that is a whole other discussion.

We have a tank for the goldfish my daughter won at a couple of county fairs over the years. Be aware that goldfish are very "dirty" - they live in colder water than many other freshwater fish you might get, they produce a lot of waste, and they can live for 15 years. Despite the unprepossessing circumstances of how we got them, several of them are still going strong, and pretty big. Because they are such dirty fish, you have to use a very large tank (we have 3 of them in a 30 gallon tank), and change at least 15% of the water every week. 

My point is that getting fish may turn out to be a bigger undertaking than you think, unless you treat them as disposable pets. Which is probably not a very good lesson for your daughter - that living things are disposable. Get the fish if you want to have the fish, not if your 3 year old wants them.

I'd also vote for the Betta. We had one that came home with us as a centerpiece from a formal event, one fish in a bowl with  a Jade plant and colored stones. The fish lived over 2 years. Survived a cat attack, too.

Just don't overfeed, kids love to feed the fish but one fish does not really eat that much!

Just a devil's advocate story here. 

I'm getting my mail the other night and a lady drives-up, walks to the community board in the mail area and starts hanging a sign offering a free Betta and tank. 

Apparently she didn't realize they eat other fish.  So if you just want the one fish, have at it.  But if you want more than 1, don't get the betta.

Well that'll be a good "life & death lesson" for her at 3.

Yeah, betas are sort of hardy, but I'll bet that within 12 months she'll suddenly find fishy "sleeping with his eyes open on his back on top of the water".

It's just the nature of the beast with fish.  We went through 3 betas, 2 gerbils and 1 mouse in our house.  I've got a real "pet cemetary" going out in the back flower bed.

To sum up:

1. If you go goldfish; they MUST be at room temperature, not heaters; they can not be with "tropical" fish that need warmer water.  So it's either all goldfish, or all tropical.

2. As a first tank setup, stick to freshwater.  Salt water looks great, but maintaining salt levels is more work, and the whole thing is more expensive, including the fish.

3. Beta fish; specifically the males, they are the ones you see in bowls by themselves; you know why?  They fight with pretty much anything else in the tank.  Note that they are actually "lung fish", in that they breathe air from the surface, so they need to be able to get to the surface occasionally.  A friend learned that the hardway, overfilling a glass vase that had a plant in it.

4. Check Craigslist/ebay/freecycle for tanks or complete setups.  You can find some good deals there, if you don't mind cleaning the equipment up a bit, if you're thinking of going with a larger tank.

LOL.  That brings back some memories.  We have about half a dozen hamsters, a bunch of fish, some turtles and a hermit crab buried out back.  We still have one hamster that's MIA, but after 15 years, I don't hold out much hope...

Back to the OP, I'd go with a couple of goldfish in a simple bowl.

Mark

I think live bearers such as guppies or mollies would be good.  As a kid, we kept guppies in a half gallon jar until they had babies, then we got a 10gal tank for them.

Betas are good, you can put them in a tank with the guppies, you just can't keep 2 males in the  same tank.  There is a reason why they are called "Siamese Fighting Fish".  

For 2 goldfish, that is very overcrowded! I would consider either changing the water more often, or biting the bullet and getting a bigger tank - at least 10 gallons - right away. To put it human terms, imagine having 4 people in a studio apartment. That is the kind of crowding you have going on right now in a 5 gallon tank.

At 13 cents, I assume they were feeder fish. But our fish were also "throw away" fish won at a carnival (so not high end koi), and are now about 4 years old and several inches long.

x1000 on the Betta.  

They cost next to nothing, they are really pretty, low maintenance and hardy.  they make great first 'pets' for young children.  

We have had 3 Bettas in the past 10 years - all named Fanny and all male (the kids picked them and named them).

They live about 2-4 years so its a good way for kids to learn about death and burial.

That's a tried and true method of rapidly cycling a tank that some aquarists use. I did it one time, and felt so guilty, I just let the remaining goldfish live until one of the cats ate them while we were on vacation.