Can a ~$300 Budget Laptop Play LOL Well?

What's the lowest price point I could play LOL well at (for a laptop)?

Don't play on a laptop. It's that simple.

By far, the best of the laptops you list is the first one you list from newegg (link ending in N82E16834314540); it's still garbage for gaming, but consider it the "best of the worst." That Lenovo is especially garbage, almost to the point of being insulting.

EDIT: Taken from your OP

primarily for school

Colleges which actually leverage the benefits of laptops usually hand them out to students, so that they can ensure any necessary software is already installed.

Unless you can specifically list a course or courses in which you'll explicitly need the laptop, then the chance that the laptop will be stolen out of your dorm room is greater than the chance that you'll ever even bring the laptop to class, let alone actually use it in class.

If you intend to play anything more intensive than a facebook game, no budget laptop will be sufficient. You're better off buying a $600-800 desktop from dell than wasting $300+ on what amounts to a glorified pr0n machine (and the ones you listed might not even be very good at offering that; I certainly wouldn't stream netflix for any length of time).

Alright homie, I'll take a minute out of my day and give you the rundown.

The two things you are looking for when trying to gauge performance is

1. The processor (the thing with 'cores')
2. The Video card

1 - The processor generally isn't an issue, most new laptops have a lot of money spent on their processor because in adverts the whole "INTEL MX 9 MILLION" is a good sale point; you want a quad core processor or more (for simplicity, more cores means better)

2 - The graphics card is the second thing you are going to want to check for when gauging performance. Most people get hung up on this because they don't understand why a Video Card (Graphics card) is important; most people see the 4GB of RAM they have and think "why do I need a 1gb video card? I already have 4GB of ram and 5GB doesn't seem that much better than 4GB."

This is a picture of 4GB of ram

http://1.bp.blogspot.com/-5xODMeFMxuQ/Ut3VNmJ7puI/AAAAAAAADvo/bgbuuX837qA/s1600/2014-01-21_090112.png

This is a picture of a 4GB video card

http://www.blogcdn.com/www.engadget.com/media/2013/10/grx780.jpg

Notice the size difference?

TL:DR

Unless you build a computer yourself, you aren't gonna be satisfied with a sub 500$ computer for playing games on. If the graphics card of the laptop you are going to buy says "intel HD" and the number is less than 4000 * *DO NOT BUY IT. ** You want a 4gb ram + quad core + a half GB video card.

Decent/passable given your budget. http://www.newegg.com/Product/Product.aspx?Item=N82E16834258152 http://www.newegg.com/Product/Product.aspx?Item=N82E16834313641

But ultimately decided I'd prefer a laptop (and then down The line, when I have more money, build a sweet desktop).

You can upgrade a budget desktop one piece at a time.

1 - The processor generally isn't an issue, most new laptops have a lot of money spent on their processor because in adverts the whole "INTEL MX 9 MILLION" is a good sale point; you want a quad core processor or more (for simplicity, more cores means better)

I highly doubt there's a budget laptop with a quad core. Most laptops use dual core processors, to the point of using dual core versions of what would be a quad core processor on a desktop.

This is a picture of 4GB of ram

That stick is plainly for a desktop.

This is a picture of a 4GB video card

That's for a desktop as well.

If the graphics card of the laptop you are going to buy says "intel HD" and the number is less than 4000 * *DO NOT BUY IT. **

Anything lower won't support dx11, but games which require dx11 probably won't run well on a budget laptop anyway, and if they do run well the laptop will probably reach too hot to comfortably touch after any reasonable amount of gameplay.

http://www.newegg.com/Product/Product.aspx?Item=N82E16834258152 http://www.newegg.com/Product/Product.aspx?Item=N82E16834313641

Neither looks very good. That HP uses a dual core i3, violating your own advice you laid out above; the Lenovo at least isn't as much a ripoff as the Lenovo the OP posted, but I would suggest any casual enduser to avoid AMD.

The one below which the OP linked earlier is better. In most regards it's more or less the same as the two you list, but with a newer i3 dual core, a newer integrated video chipset, and its motherboard can support up to 8GB of RAM (I didn't bother checking that for the ones you listed).

None of them seem to offer expansion slots for an independent video card.

http://www.newegg.com/Product/Product.aspx?Item=N82E16834314540