The same standard 6 ability scores have been with D&D since it's inception and they have gone on to impact the entire RPG hobby, but why not change up ability scores and what they mean in a campaign? If a DM pays attention to how ability scores work within the rules of the game they are using those ability score themselves can be changed to provide a different game experience.
No ability scores at all. The most drastic means of addressing the situation is to simply not have any ability scores. In the original D&D rules ability score had little direct impact on most immediate actions so little actually some have wondered why they were bothered with at all. By ditching abilitry score entierly all adventurers are equally capable at common adventuring tasks and no player is slave to bad luck and no one ends up subordinate to the early lucky rolls of another player. This method is used by some minimalist D&D variants out there “Searchers of the Unkown” being the one that come to mind first. Characters abilities are tied to class, level, and player choice a character will emerge from play instead of being rigidly defined in a few small areas. There is room for this method to be exploited by pushy and intelligent players that may cause it to break down for a wide open and extended campaign but in more tightly focused campaigns or limited duration ones the ability scores can be an unneeded layer of complexity.
Ability scores as modifiers. I’ve use this technique in the past when playing Talislanta and for a little quick and simple multi-edition D&D dungeon crawling as mentioned in this post: http://aeonsnaugauries.blogspot.com/2011/10/weekend-gaming.html
Ability score as modifier has two particular advantages over the standard score and modifier model in that they are unbound, and concise. Both advantages can also be a deterrent in their use however. Ability score as modifier is concise as there is no look up required to determine the effect of an ability on a relevant action if you have a +3 in Might you have a +3 bonus to all things might related. An ability score as modifier is unbound you can technically go to infinity in either direction as there is no score floor or ceiling implied by this method; you don’t’ have to have a housecat with a strength of 3 (or 1). There is a problem in the ability score as modifier as it isn’t clear how relatively effective or weak any two characters really are in a game without really looking into the numbers (a +4 is cool if you are roll 1d20 and adding 4 but it’s even better if you are rolling 3d6 and adding) and some baselines have to still be established which can look odd compared to a ability score as modifier mechanic.
Changing the Names of ability Scores. Using the same core set of modifiers by changing the definitions of the ability scores. Why Strength, Intelligence, Wisdom, Dexterity, Constitution, and Charisma? I explored this option in a post a couple years ago in this post: http://aeonsnaugauries.blogspot.com/2009/10/whats-in-name-or-ability-scores-and.html . In brief the game may feel different by simply changing a few names and the expectations that go with them as a result while otherwise retaining existing modifiers. By example: if Strength is instead Size some the descriptions of things certainly shift while still playing out effectively the same numerically in uch of play but there will be occasional telling differences; big folk can use big weapons better, they can surely bash open door easier but what happens when high Size characters have to squeeze through a small hole?
Additional ability scores. Why stick with just the ones your game came with? Even AD&D added Comeliness to the standard roster of ability scores for a time why not consider doing the same in your campaign? Core tasks repeated again and again in the game might work better if tied to a new ability score; Perception as an ability score is a popular one that came up in Dragon magazine ages ago and is a popular candidate. Luck is a popular ability scores used by some D&D-a-likes that would fit in well as a modifier to the saving throw system or perhaps a complete alternative. If characters are going to spend a lot of time on horseback in a campaign and the DM feels there should be some difference between how well characters ride it might be worth adding Horsemanship as an ability score as opposed to adding on a skill system. One could even delve deeper into some of the functionality of ability scores now and break them down further sure a high intelligence makes one a better magi-user but what if magic was tied to something else maybe Magical Affinity with minimum scores establishing spell access and/or magic item use instead of the intellect alone? How about breaking up that Charisma score into Looks, Leadership, and Bargaining surely that’s going to lead to a wider range of play and more varied characters.
Fewer ability scores is an option as well one could reign in the number of ability scores to more generalize them to have more evocative relationships t common areas in a campaign. For example: Strength and Constitution get rolled up together in Vigor, Dexterity and Intelligence get wound up into Craft and Wisdom and Charisma are wound up into Spirit; maybe some of the modifiers are dropped, maybe they are just carried along, it’s up to the campaign of course. With such a scheme the strong (unless diseased or injured) are usually also healthy, the wise know the right thing to say and more readily influence others than could a rude oaf with a charming smile.
There is no need to leave ourselves tethered to the same standard six ability scores in our own campaigns, the rules as written are a starting point not the be all and end all of the possibilities in a campaign.