The standard grips on contemporary Odyssey putters are all manufactured by the Golf Pride company. Golf Pride has been the top seller of golf grips for generations and it produces a wide variety of grips for an even wider variety of club manufacturers. Most of the current standard grips are a leather-style rubber grip in colors that match the club design. The grip for the original White Hot series is still exceedingly popular; it is a black rubber grip with white textured hash marks.
There are about numerous different grip styles branded specifically for Odyssey putters. Most of these are in the leather-style soft rubber grip family. Soft in texture and colored to match the different Odyssey brand models, these putters have a satisfying feel but can get slippery in wet weather.
The following grips are available through Odyssey:
Super Stroke Odyssey: These come in 3 sizes and are styled to match the model of putter, including the Big T, White Hot Flatso, Tribecca, among others. In addition, there are the slim White Hot Pro .580 and the oversized Jumbo Lite models.
The above-named grips are the types of grips that come standard on new Odyssey putters. If your grip wears out or becomes slick, you can certainly find an exact replacement for the grip on your putter or replace it with any one of these--or any other putter grip, for that matter. Tiger Woods famously used a Ping putter grip on his Titleist Scotty Cameron putter. It is most important to have a grip that feels comfortable to you on your own Odyssey putter.