Kids are definitely a challenge but then some adults are also. Games with essential pieces (e.g. Clue, Monopoly, puzzles and any card game that doesn't use a standard deck) are a "lost piece" problem and I don't provide them.
Mancala is a game that amazingly young children can play... and win at. The board is an attention getter as many people are not familiar with it (I leave it out with a small card explaining the rules.) The pieces are polished/smooth stones and/or dried beans- which are easily replaced as they do not need to match. Set this up high enough to avoid the really young "taste everything" crowd.
Treasure hunt. For younger kids I made up a one page check list with drawings of things they can find in the rental which requires them to explore the entire place (rooms, floors and ceilings) to find them and check them off the list. Obviously this is aimed at non-readers.
Incomplete card decks. I mark with an X on the back and keep in a "building box" for older kids (with more dexterity) to build as big or tall a card house as they can. Ask them to email you a photo. Adults get into this one. If they grab a real deck it is easy to separate because of the X.
Paper and pencils. Besides keeping score if required, I add a suggestion/hint list of games on top of the box as sometimes a bored child just needs a little prodding to remember. Hangman, Tic-tac-toe (naughts and crosses), Sprouts, battleship plus many more for those that can spell- then there is just drawing. I no longer provide any crayolas, markers, paint or craft materials but I do have an old messed up "craft and game" table cloth.
A roll of masking tape (narrow and really low glue strength- I actually use that blue painters tape) can allow hop-scotch on the floor, "balancing beams"- I've never had it abused (put on floor only) but there is extra clean-up involved.
All too often merely making suggestions is all that is needed. I made a short list of games that do not require pieces, parts or boards like "I Spy", "Hinky-Pinky" , "Twenty questions", etc. Admittedly, I haven't finished compiling rules/directions for these.
I keep extra die pieces in a labeled plastic jar and don't worry if they don't match.
For games that only need player pieces (not miscellaneous essential parts like Clue) when a player piece goes missing I just provide objects that can be substituted - thimble, key, unique rock, washers, bolt/nut, foreign coin, etc. Another labeled jar.
I use labeled, clear plastic boxes with lids to individually store these items. Large enough to allow things to be thrown in them and not requiring being "fitted"- too time consuming and it never happens.
What has been your experience with where to put this stuff? I have tried down low where the child can see and access them, but it seems they all get dragged out and scattered about. Storing them up high requiring an adult or older kid to reach them seems to result in their never being used.