A few games come to mind as readily modifiable for blind play, and a few as playable out of the box.
Out of the box
Versions of Othello have been released in the past with raised lines defining the spaces; the pieces in that are slightly concave on the black side, and flat on the white. Othello Classic appears to be the current title for this, but I can't tell if it has the standard pieces. (Note that the Travel version has double-flat pieces, while many versions of the full size have simple printed grids, not raised.) Maxiaids.com has a version using rods with a brad in one end, rather than discs.
Mancala as a 2-player game can be played out of the box with only minor issues - the typical rule of not touching others' pits must be ignored for the blind player. Mancala as a series game, as presented in "capture the pit" play in the Oh-wah-ree version by 3M, has issues in multi-player (marbles being used to denote who owns which pit), but one player could substitute beads, another use empty pits, etc, allowing for even multi-player play.
Dominoes. While a traditional "family game", it's actually got some strategy, and is inherently blind-friendly with standard pieces. One blind acquaintance has burned out on it because it was the only blind-friendly game at school.
Rumikub. Some sets are deeply embossed; with one of these, the blind player may be able to feel the numbers without modifications. We had to go to using face down pile on the table instead of drawing from a bag because my set was deeply enough embossed for me to tell what I was drawing.
Mah Jongg can be played once the blind player learns the piece faces - a larger set is better. Most sets use engraved images (the others are cards). Since even the scoring is done using dot-marked sticks, it's just a matter of learning curve.
Score Sheet modifications only
Catan Dice - the dice are only slightly embossed in the wood dice versions, but more deeply so in the plastic dice versions. The score sheet can be replaced with a braille linear matt, and markers for showing what's been taken. Coins can be used for score keeping.
Modifications - add texture
Travel Blokus, with the smooth pieces, can be modified by careful drilling. Drill holes through one color, in every square of every piece of that color. You now have a tactile version. The blind player can feel the board state, the pieces aren't likely to move, and there is no information loss nor gain.
Standard Blokus - if you have two identically sized sets, one with the pieces with hollows, and one with the smooth sided pieces, you can swap a color out in the process. If you have a single smooth-faced set, drill out one color as for travel. If you have a hollowed set, fill the hollows with glue. Or use a texturing on one color. The blind player can feel the alternate face, but can't ID the other colors as separate, but their own move is only trivially hampered by playing this way.
Hive - Since the pieces are not randomized at all, and the images are depressed, only a tiny modification is needed - some means of telling one color from the other; texturing is suggested. It would be best to play on a slightly tacky surface, such as a warm vinyl placemat or a non-slip cabinet liner, so the pieces don't readily slide when touched.
Kids of Catan - you need to texture the player pieces, but that's it. The die is pipped, the commodities can be told by shape, and one can readily play ignoring the slot colors (and in fact, doing so makes the game far more tactical, as does allowing movement either direction). One could add braille labels to the slots, as well, and play the game unmodified. It's simple, but with allowing movement either direction, not totally inane, and very pretty for the sighted folks.
Cathedral, as a game, should be playable, except that the pieces don't lock to the board. This can be fixed with drilling holes in the board, and adding not-quite-flush nails to the piece bottoms to lock them in. This also turns it into a decent back-seat-of-the-car or on-the-bus version for longer trips. for totally sightless play, one should texture one set of pieces, as well - sand glued to the roofs is practical and pretty.
Chess can be modded by use of a board with raised square edges and two different sets of pieces with different shapes, and again, ignoring the no-touch rule. Alternatively, the board can be drilled, and pins inserted into the pieces' bases, and one side textured, just like Cathedral.
Diplomacy - the hardest part is not knocking the pieces around. The solution? use a thin bead of caulk to define the board space's edges, braille label the spaces by name, and use distinctive shaped or textured pieces for each player. One might also texture (with blue sand) the seas. For example, one player might be full height cubes, another half height, a third half height textured, a 4th using cylinders full height, and 5th with half-height cylinders, and a 6th and 7th using full and half-height triangles.
Many games pieces can be textured. For games like Hive, where the pieces are bakelite, or blokus, where they are polycarbonate, and shape determines piece type, one can texture the surface with only minor cosmetic issues.
If you have a belt sander available, use a medium grit belt and a VERY light touch. Using double stick tape, secure the pieces to a piece of scrap board, and clamp that to the table. Make a single pass with the belt sander on low speed in order to create furrows on the surface. Flip, and repeat for the other side for blokus pieces.
If you wish to go the glue and sand route, apply a very thin layer of glue to the face to be textured, then press that face into a bowl of sand, remove, and let dry.
A number of games using bakelite or plastic tiles can have braille drilled in; some blind folks find this acceptable, others don't. If used, set your stops on a drill press to prevent drilling completely through, and mark all the dots first.
Brailling of cards
Most card games can be done in braille - either by directing brailling the cards, or by braille stickering.
A few have been done by punching holes in the cards, but this marks the cards.