Phoebe

I thought that it would be fun to look at different pets that don't get a lot of attention these days.

You have a few main options that are low maintenance but still nice pets. Low Maintenance doesn't mean low cost tho.

Fish

Cheapest, low cost option would be get a 5 gallon tank, heater, some timed lights, and get a beta, with a snail or two. You will need to cycle the tank (look up the nitrogen cycle, and a lot of you tube vids and guides on how to do this, but the basic of it put some fish flakes in an tank with out fish and let it sit for 2 weeks) Then keep the water around 70ish, and then you only need to clean the tank once every week (taking about 30% of the water out and replacing it and adding a dechlorinator, and quick wipe down of the tank walls), and feed daily. The snails will help keep the tank algae free.

Cost about a $100 bucks to set up.

These are fun to look at but not much else.

You'll spend about $5 bucks a month (food cost as well as supplies)

Spider

More costly then a fish, as well as still a pet you watch not play with, they cost anywhere from $15 on up. I would suggest a Rose hair or a Golden Knee as a starter. You will need to set em up with a terrarium , heater , and keep it semi moist, as well as you will have to feed em crickets or grubs so if live insets creep you out stay away. Clean cage weekly, feed them 1 or 2 times a week.

Cost to set up : $200 ish for a good set up don't want to go cheap and the terrarium and have them escape on you.

While you can handle them its not advised to do it. They are to observe for the most part. Some can get rather mean when held as well and can very fragile, a fall of lest then a foot can be fatal for them.

You will spend about $10 bucks a month give or take for crickets might be cheaper where you live or if you decide to buy in bulk, but my local pet smart has a 20 pack for $2.50 for sale that perfect for a weeks worth of food.

Frogs, and other amphibians

Stay away they might seem cheap but very hard to keep alive with out keeping the right water and environment for them.

Rats, and other rodents (hamster, and such)*

You said you didn't want a Rat in your post. I would agree to, they are not very low upkeep as they are social pets, if you don't plan on playing with them daily they will grow sad and depressed. As well as you have to get more then 1 for them to thrive. Other then Rats, most other rodents are just for watch not play as well as they get stressed easy.

I would honest go for the Beta fish.