In regard to clothing, "stays" provide structure or stabilization to the fabric.  In regard to dress shirts, the stay helps the collar "stay" put.  Meaning stay in place and flat.

Collar stays help provide structure or support to the collar of shirts.  A good example would be the use in some types of dress shirt.  Note: generally not used in the button down color dress shirts.

• A collar stay, collar stick, collar tab, collar stiffener, or collar stiff is a shirt accessory consisting of a smooth strip of rigid material, rounded at one end and pointed at the other, inserted into specially made pockets on the underside of a shirt collar to stabilize the collar's points. The stays ensure that the collar lies flat against the collarbone, looking crisp and remaining in the correct place.
• Collar stays can be made from a variety of materials, including metal (such as brass, stainless steel, or sterling silver), horn, baleen, mother of pearl, or plastic. Shirts often come with plastic stays that may eventually need to be replaced if they bend; metal replacements are less likely to have this problem.
• Collar stays are removed from shirts before dry cleaning or pressing.

Learn more about shirt collars.

In corsetry, a bone works similar to a stay in the fact that it provides support or stabilization to the garment. 

• The bone is one of the rigid parts of a corset that forms its frame and gives it rigidity. The purpose of the boning in a corset varies slightly from era to era. Generally, the cinching/shaping properties of corsetry puts strain onto the fabric from which the corset is made. The boning supports the desired shape and prevents wrinkling of the corset fabric. Bones, and the substances used for the purpose, are generically called "boning"; however, the name likely arises from the use of whalebone in early corsets. Today, many corsets use nylon or Rigilene boning, although steel is still favoured for high-quality corsets. Plastic bones do not have the strength required for tightlacing and are known to warp and bend, often in unflattering ways. Many modern bodices, strapless gowns and lingerie use a variety of plastic boning because it is much cheaper. Modern steel bones come in two basic varieties: "flat" steel boning (sometimes called just "enamel" or "rigid" boning) and "spiral" steel boning. Spiral boning is flattish but thicker than flat boning because of the tips required on the ends. Flat boning bends in only one direction, while spiral steel boning bends easily in two directions. Spiral steel boning may thus be used on curved channels or where a more gentle support is acceptable.