Reddit mentions of Lee's Mini Economy Gravel Vacuum

Yes, you can do a fish-in cycle with a betta! It is not ideal, but he should be fine as long as you do your proper water maintenance. Get yourself a liquid (not paper) water testing kit and do daily water tests for ammonia/nitrite/nitrate and perform a partial water change (~20% or so) whenever they are detectable. This will likely be a few times a week. As your nitrifying bacteria establish, you'll be able to go for longer periods of time without doing those water changes. When fully cycled, you should do just weekly partial water changes. Seachem Prime is a great brand of water conditioner, as it's extremely concentrated (a drop or two will treat a gallon of water) and has the added benefit of detoxifying ammonia for 24 hours.

If you don't have one yet, a gravel vacuum is your best friend for water changes. Just have a bucket or something to drain the water into.

I'm not sure what kind of filter the kit comes with, but I recommend scrapping cartridges and filling the filter with good biological media (like aquarium sponge, filter floss, ceramic noodles). These media give a huge amount of surface area for your cycling bacteria to colonize, and never need to be replaced. You may also want to get a pre-filter sponge to slip over the intake to protect his fins from getting sucked in.

It will be okay! You can do it!

Here's what you'll need:

• 5+ gallon tank with a lid. You can go for one of the nicer-looking (but more expensive ones) or just a plain ol' tank from any pet store. Lids are necessary because bettas are jumpy by nature.
  
  
• Heater. Bettas are tropical fish so you need something that can keep the water at around 78ºF. I'd really reccomend an adjustable one because A) you can do extra fine-tuning and B) you can bump the temperature up if you need to (if the room gets extra cold, or if your betta gets stick, etc). Here's what I use in my 5.5g.
  
  
• Thermometer. Nothing fancy, but you need something so you can know what the temperature is. Get an internal one instead of the ones that stick on the front of the glass (those aren't very accurate). Again, you can find them at any pet store for a couple bucks.
  
  
• Filter. Filters are a must because they house the bacteria that maintain a tank's cycle (preventing your fish from getting poisoned). I would reccomend either reading up on the nitrogen cycle yourself and teaching your friend or giving them a link to a guide, since if they've got the fish in a little container they probably know nothing about it. An air pump + sponge filter is a cheap way to do it but as long as you can get a filter that makes less of a current you're good. Bettas (especially ones with long heavy fins) don't like fast flowing water. I've got this one in my 5.5g. The fact that it's adjustable is super convenient.
  
  
• Substrate. Looks nice, add extra surface area for more bacteria to grow, A+. You can get either gravel or sand. If you want to go cheap, regular pool filter sand or black diamond blasting sand will get you a ton for a few bucks.
  
  
• Decor. Bettas appreciate densely planted/decorated tanks with lots of hiding places. Make sure anything you get isn't rough/sharp enough to snag panty hose, because that means it'll also tear betta fins. That means no coarse decorations, plastic plants, etc. Silk plants are popular, and mugs are an easy way to add a little cave.
  
  
• Gravel vac. When you're doing water changes you need a gravel vacuum to clean down in the gravel. I've found that this one is a great size for my 5.5g. Others I've tried pull out water too fast to get a good cleaning in before you've removed the water you need to. She'll probably need some container to put the water into as well. I just use a plastic 1-gallon pitcher for my 5.5.
  
  
• Test kit. Back to cycling, you need to have a test kit so you can know the pH, concentration of ammonia, concentration of nitrite, and concentration of nitrate in the tank. The API freshwater master test kit is far and away the most popular since it provides all four. Whatever you get, go for the liquid kits rather than the strips because strips aren't very accurate.
  
  
• Betta food, but she probably already has that. Although if she's got flakes it'd be a good idea to move her over to pellets, since flakes make the water a lot dirtier. Also I would reccomend you advise her on how much she should be feeding her betta. They're little piggies and will eat themselves sick if you let them, so people who don't research betta care are prone to overfeeding.
  
  
• Water conditioner. Water conditioner removes chlorine from tap water so fish can live in it. Oftentimes you'll see betta-specific water conditioner at stores, but this is just a scam that preys on folks who don't know better. It's overpriced, comes in tiny bottles, and is watered down. At 1/10 mL per gallon, a single bottle of Prime for instance is way cheaper and will last way longer. I'd reccomend you also get a 1mL syringe for easy dosing. Whenever I want one I pick one up from my school's chemistry stockroom for like 25 cents.