This is a question that should probably be asked more frequently than it currently is. Many current D&D players have recently grown accustomed to assuming that their characters can freely design new magic items not appearing in the core rule books, and procure such items by simply spending a certain amount of gold. This is borne out by the abundance of "price this magic item" threads on a number of D&D messageboards (which properly belong under "house rules", not core rules, discussions) these days.
The proper answer is: "No. The core rules do not automatically allow players to design new items." It's the wise (and sane) DM who responds to such player requests by saying "No" in the general case, and only making an exception if the DM him- or herself has a compelling reason to do so in their own campaign.
What's In the Rules
The root of the problem is that the Dungeon Master's Guide included a table for "Calculating Magic Item Gold Piece Values" (DMG ch. 8: "Creating Magic Items"). Many players have taken note of this table, and come to assume that it's a shopping list for their characters and item-creators to select from, which is incorrect. The table is in the DMG for a reason, namely that it's for use by DMs only for items that they add to their campaigns. It's precisely this kind of misunderstanding that motivated Monte Cook, attributed as author of that section of the DMG, to write, "Some days I look at Table 8-40 on page 242 of the DMG and wish it wasn't there at all." (here: http://www.montecook.com/arch_dmonly3.html#marketvalue ).
Consider also the following:
- The notes to the table in question, in a sidebar on DMG p. 243, begin by saying "Many factors must be considered when determining the price of magic items you invent". The "you" here refers to the reader of the book, that is, the DM.
- The sidebar ends by saying "The formulas only provide a starting point... items require at least some DM judgement calls." That is, the table is not a shopping list for players, but rather a starting tool for DMs.
- On the one hand, people making potions, scrolls, and wands can clearly put any appropriate spell in them, and use the fixed pricing rules that do appear on PHB p. 78 and DMG p. 244-6. But other item creation feats all say something like this: "See the Dungeon Master's Guide for descriptions of magic weapons, armor, and shields, the prerequisites associated with each one, and prices of their features." Merely taking such a feat does not immediately allow one to invent "new items"; they must, according to the core rules, choose from the existing "descriptions" in the DMG.
There's only one single place in all of the core rules that even refers to PCs possibly creating new magic items. The section specifically on "New Magic Items" (DMG p. 178) says the following: "In the same way that you can invent new spells and monsters for your campaign, you can invent new magic items. In the same way that a PC spellcaster can research a new spell, a PC may be able to invent a new kind of magic item." (Emphasis mine.) Again, "you" here refers to DMs, not players. Just as players do not get to invent new monsters or spells (without research), they do not by default get to invent new magic items. The passage says that PCs "may" be able to create new magic items, if so allowed by a generous DM, and it refers back to the new-spell research guidelines in order to do so.
What Gets Broken
Let's say you're a DM that does in fact allow players to use DMG Table 8-40 as a shopping list for new items of their own specification. Then there are a number of D&D balancing issue that suddenly get broken to a severe degree:
1. Playtested Magic Items. D&D 3rd Edition had an impressive playtest program, with quite a bit of input and history behind the items in the core rules, and how they work in conjunction with the rest of the system. Clearly Table 8-40 was not itself playtested, nor were any items invented by your players. It's easy to create broken items by blindly following that table, and so the DM must either be allowing broken items or on-the-fly be making balancing judgements of lower quality than the rest of the playtested rules.
For example: One entry suggests that a "Skill bonus" be priced at the "Bonus squared x 20 gp". If any such skill bonus item were available, economically, everyone should be able to afford and purchase a +10 (or more) skill item in their key profession, from Craft (blacksmithing) to Use Magic Device and Spellcraft. In fact, they look stupid if they fail to do so. However, allowing this in a campaign breaks all the mechanics behind Craft, Use Magic Device, and Spellcraft (esp., the ELH's spell-seeds system), because such items did not exist and were not used for either PCs or NPCs in the playtest of the D&D rule system.
2. The Item Creation Feats. By the core rules, a separate item-creation feat is required for each category of magic item (arms and armor, potions, scrolls, wands, rods, staffs, rings, and wondrous items). However, if anyone can arbitrarily design their own items, then any item creator only needs to take one of these feats. A wizard can just take Craft Wondrous Item and then invent a series of new items that incorporate all the powers from any other item category in the DMG that they desire. Obviously, the whole item-creation feat system breaks done totally as a result.
For example: A great many D&D players express an honest bewilderment at why the multiple item-creation feats exist in D&D, when they can just specify their own items. They think that feats like Craft Rod and Forge Ring are "worthless". Indeed, they are absolutely correct, in any campaign where new items are freely invented. However, if the DM sticks to the items described in the core rules, then suddenly the fact that powers such as elemental resistance, invisibility, deflection bonuses, and spell turning can only be obtained via Forge Ring, makes that feat looks very valuable indeed, and the sensibility of the core rules as written is clarified.
3. The Space Limit System. The core rules provide a list of a dozen available spaces (slots) by which a character is limited in the magic items they can usefully carry (DMG ch. 8: "Limits on Magic Items Worn"). But again, this system breaks down if a player can always invent an item that does whatever they want and place it in any arbitrary space (or worse, make the item take up no space or invent their own space type). The system instead gets effectively replaced by one of "you can carry up to twelve items (before paying extra gold for the privilege)", and the "slots" become meaningless.
For example: The DMG p. 176 says that "if a character puts on another magic cloak on top of the one he is already wearing, the second cloak's power does not work". Thus, a PC must choose between a cloak of charisma and a cloak of elvenkind, and cannot use both at the same time. However, this becomes meaningless if the player is allowed to sell one of the cloaks and buy a hat that does the same thing, at the same price. A primary balancing factor and interesting in-game strategic choice has thus been removed from the system.
There are a great many suggestions, options, and variant rules provided in the Dungeon Master's Guide. However, these are all provided for DMs as options in customizing their game campaigns. It's erroneous to think that these possible modifications count as "core rules" (especially when they do not appear in the SRD) or can be demanded by players. For example, the option of "researching original spells" is given in a list of variant rules (DMG p. 42 and 95), but strict oversight is given to the DM, along with weeks-long research times and thousand-gold-piece expenses ("new item" creation specifically references back to these rules). To be balanced, new item creation (which only in passing even suggests the possibility that PCs can do it) needs to be at least as intensive a process as new spell creation.
If a DM has any doubts that a player inventing a new magic item is appropriate in his or her campaign, that DM should feel entirely confident that an answer of "No" is the most appropriate, simplest, shortest, and assuredly well-balanced answer that they can make. Players shouldn't get to invent and graft new abilities for free onto their characters (be that race or class abilities, feats, spells, or magic items) - that's the job of the DM, to provide the campaign setting and the options from which to choose. Character abilities are only truly interesting when they come as a result of a choice which excludes some other possible advantage.