Is that $100 to $200 total or per player? Either way it may be a lot more than you need to spend.
Yes, there are a ton of MP3 players. Prices range from $10 or so up to several hundred dollars for an iPod touch or a player based on a hard drive rather than flash memory (which, by the way, I would AVOID LIKE THE PLAGUE (regardless of brand)). Most modern MP3 players use flash memory, but a few (very expensive models) use actual hard drives. Hard drives have tremendous capacity, but they use a lot of power, they are F-R-A-G-I-L-E and in my mind they no longer belong in MP3 players at all. So that is the first consideration, but there are very few hard drive based players still around and they are very expensive, so they would probably rule themselves out anyway.
Here are some more criteria to consider:
Memory size: If you get a flash based player, how much memory do you need? The choice here is from about 1 Gigabyte to 8 or more Gigibytes. One gigabyte will hold about 300 to 500 songs, give or take. In truth, if the player is being used ONLY for music, 1GB is enough for most people, but in today's world .... more is better and the incremental cost of another gigabyte or two is only $5 to $20.
Memory expandability: SOME players (very few, actually, but some) have a memory card slot for adding memory just by plugging in a digital camera flash memory card.
Video Support: Notice that in the previous paragraph I said "if the player is being used only for music" ..... well, many people put not only music (MP3's) but also video on their MP3 players. Not all MP3 players support this, but a lot of them do once you get above the low-end products. Do you want this? [You probably do]. Be aware that once you start supporting video, your memory requirements and the need for a far fancier interface and display go way up. But, in today's world, the cost can still be surprisingly reasonable.
Display screen: Some low-end MP3 players have only a very simple display screen (black and white, mostly just a line or two of text ... not much more of a display than a wristwatch), while more advanced units have full-color screens varying from just over an inch to as much as 4" or more. Obviously, if you are going to buy a unit that supports playing video, it MUST have a full color screen.
Battery type: There are 4 basic options for the type of battery used, and in my mind this is an important characteristic. Some players use standard AA or AAA batteries (either disposable or rechargeable) (this is actually good, but it makes the players BIG). Some players use proprietary, model specific lithium rechargeable batteries that are user replaceable. Some players use proprietary, model specific lithium rechargeable batteries and when the batteries die (and they WILL die, in 6 months to about 3 years or so) you have to send the player back to the manufacturer to have them replaced (and, in some cases, YOU WILL LOSE YOUR STORED MUSIC WHEN THE BATTERY IS REPLACED). And, finally (this one is incredible but true) some players use proprietary, model specific lithium rechargeable batteries and when the batteries die you are expected to throw out the entire MP3 player and buy a new one; there is NO battery replacement available at all.
Other functions: Many MP3 players have built in FM stereo radios as well (SOME of them can also make recordings, in the device, of whatever is currently being broadcast on the radio). Many MP3 players have a built-in voice recorder. Many MP3 players can also store and display, in addition to music and video, still photos. Some MP3 players have built in or optional games (Tetris, Pac-Man, etc.)
Music Service and Music Format compatability: Players vary widely in how you get music into them, and in how they dowload and play music files that are not MP3 files but rather some other format that contains "DRM" ("digital rights management", aka copy protection). For example, the iPod line, while it will work in some other ways, normally expects you to install "iTunes" on your computer and it's primary intended music format is not MP3 (which, however, it WILL play) but rather Apple's "AAC" format that has DRM. Other players similarly are linked to other Music services such as, for example, "Rhapsody". But it's nice to have a player that does not require the use of either ANY software or ANY music service, that just looks like a "disk drive" (in "My Computer") when you connect it and you put music onto it by just using "drag and drop" from your computer in "My Computer" or Windows Explorer (this is sometimes called "MSC" mode). Another kind of generic option is players that use Windows Media player as a "generic" music management program (Microsofts intent was to support DRM-enabled WMA files, but it works with non-DRM MP3 files also), this mode is called "MTP" mode.
Let me first mention the one reason why, while it's not my favorite MP3 player, you possibly should get an iPod (the overall brand name for all of the MP3 players made by Apple computer): Accessory availability. There are a ton of iPod accessories, and very few for any other models.
Now, having said all of that, a specific recommendation:
I've been working with MP3s for over a decade, and I have to tell you that my favorite MP3 player is the Sandisk Sansa e200 series (available in 4 memory sizes from 2GB to 8GB as the e250 to e280 models). This is a wonderful player, in my opinion it has the best features of every single charateristic that I described above. It plays Music, Video, still photos, it has an FM Radio (which it can record), a voice recorder, it has both MSC and MTP modes, it has a user replaceable battery and it has a memory card memory expansion slot.
Now for the bad news: The e200's are discontinued !! They have been replaced by the Sandisk Sansa Clip (no video support), Fuze (nearly an exact replacement for the e200) and View (optimized for video, has a larger display screen). But I don't like these as well as the e200 series (really, the e250(2GB), e260(4GB), e270(6GB) and the e280(8GB)) becuase when their batteries die, you are expected to throw out the entire player and get a new one ... incredibly, there is NO battery replacement for these newer players, PERIOD.]
[Just as an aside, on my e260 (4GB), I have 404 songs, 3 short videos (2 are "American Idol" performances that I taped on a VCR and then put on the computer) and 172 still photos .... all in about 800MB (0.8GB). The player is less than 25% full (in fact, all of this would fit on a 2GB e250 and even it would still be more than half empty).]
But hark [the Herald Angels Sing]!! All is not lost.
There are a LOT of "refurbished" e200's available and they are CHEAP !!! I mean, generally, $25 for the e250's up to $49 for the e280's. Keep in mind, these are fantastic, high-end players .... a "new" e280, if you can find one (and you probably still can, if you look) is $199 to $249, but you can get a perfectly good "refurbished" one for about $49. And they are WIDELY available. What's more, it is reported that these "refurb" units actually ARE new (e.g. no one else ever owned them). Reportedly, they are actually not "refurbished" but rather "overstock" units that were left over when the Clip, Fuze and View replaced the e200's (I can't verify this ... but I've bought 4 of these, and this information LOOKS accurate based on the condition of the players I received).
If you are interested in an e200, you can get them on E-Bay (of course, you can get ANYTHING on E-Bay) but also at BUY.COM, TIGERDIRECT.COM, COMPUSA.COM, NEWEGG.COM and AMAZON.COM, although they often (VERY often) sell out, disappear and later reappear again; availablity is sporatic. [If you are familiar with Woot.com, they also often have them about once a month.] The e250 through e280 models are absolutely identical except for the amount of memory that they have. Note that the "refurbished" models do say "refurbished" on their backs (while it looks engraved, the raised letters are actually adhesive stickers that can (with difficulty) be removed). Also, the "refurbished" models do not come with all of the accessories that a new, retail packaged model came with, but they do come with the essentials (but note that you will have to download the manual (as a PDF file) and the video/photo transfer program from the Sandisk Sansa web site). Do be sure that there is some warranty with the player that you get, because there have been a number of problems with the headphone jacks of some units having been poorly installed and not working reliably.
I hope that this has been helpful. A trip to any large electronics retailer will likely let you see probably dozens of models of MP3 players on display that you can examine and ask questions about (you generally won't find an e200, but they are similar to the Fuze models). I do think that an MP3 player will make a truly excellent gift for any teenager (or Adult!! ... get one for yourself) who does not have one, and it's not necessary to spend a lot of money on this to have a truly great item.
Hope that this helps.