There are a few different options, depending on how permissive the DM is being. Magic items and levelling in a given class are the predominant options.
At the end of this post I've included all of the various options currently available in D&D to increase your ASI (not including options in published adventures...because spoilers). I've also divided these into Permanent and Temporary designations. In dividing the increases between Permanent and Temporary I've designated anything which only increases the ability score while you are able to use the item/is as a result of a dispellable spell as "temporary".
Further to this I've divided the Magic Items below into both Major/Minor and Rarity designations. All Artifacts should be considered above Legendary items (meaning it should be super rare for any adventuring party to find these, even at Tier 4!).
Xanathar's Guide to Everything has a suggested number of magic items that a party should get, divided by tier, rarity and whether or not they are major/minor magic items (Potions included). [XGtE Pgs 135, 140 - 145]
The relevant Magic Items for ASIs fall into these ranges:
Tier 1: XGtE suggests that a party will find 1 Uncommon Major Magic Item and 1 Rare Minor Magic Item.
Tier 2: XGtE suggests that a party will find 5 Uncommon Major Magic Items, 1 Rare Major Magic Item, 5 Rare Minor Magic Items and 1 Very Rare Minor Magic Item.
Tier 3: XGtE suggests that a party will find 1 Uncommon Major Magic Item, 2 Rare Major Magic Items, 2 Very Rare Major Magic Items, 1 Legendary Major Magic Item, 9 Rare Minor Magic Items, 5 Very Rare Minor Magic Items and 1 Legendary Minor Magic Item.
Tier 4: XGtE suggests that a party will find 1 Rare Major Magic Item, 2 Very Rare Major Magic Items, 3 Legendary Major Magic Items, 4 Rare Minor Magic Items, 9 Very Rare Minor Magic Items and 6 Legendary Minor Magic Item.
We can use these guides to give us some idea of what will be appropriate from the other options we have.
Tier 2 specific info
For low Tier 2 we can expect a party to have 1 Uncommon Major Magic Item and 1 Rare Minor Magic Item.
For high Tier 2 we can expect a party to have 6 Uncommon Major Magic Items, 1 Rare Major Magic Item, 6 Rare Minor Magic Items and 1 Very Rare Minor Magic Item.
Using these ranges we can expect the party only to have 1 Magic Item that increases a stat permenantly (Bag of Beans), 6 Magic Items that increases a Strength/Intelligence to 19 while the item is worn or which temporarily increases their Strength to 21, and 1 Magic Item which will either give you +2 Constitution, set Strength to 21, or set Constitution to 19 while worn, or which temporarily sets Strength to 23/25.
All of the worn items require attunement.
The +2 to Constitution with a Maximum of 20 is equivalent to an ASI. The ones which set a stat to 19 can be quivalent to one or more ASIs (depending on what the base stat starts at).
Thus we can probably give one party member something like an ASI/feat.
As a result our Tier 2 options are:
- Do nothing and let the character use their level 8 ASI from their ordinary class progression (assuming no multiclassing)
- Give the party a relevant magic item
- Allow the character to train a Feat [PHB] & [XGtE]
- Give a member of the party a Blessing from a Deity or a Supernatural entity (which you can choose)
- Get lycanthropy or vampirism from coming across a monster that has this template applies
Training a Feat
For gaining feats we get some guidance in the Treasure section of the Dungeon Master's Guide [DMG Pg 231]. This suggests a feat is an appropriate reward for a special type of training, and that it would use up some/all of the characters downtime.
XGtE gives us some guidance on how long this training might take. It suggests that learning a language would typically take 10 workweeks, reduced by an appropriate number of workweeks equal to the characters Intelligence modifier (0 - 5), but doesn't get increased by a negative modifier. It suggests a cost of 25gp per workweek (if we are hiring a trainer as opposed to getting it rewarded to them). Every 10 workweeks spent training bring a chance of a complication occurring (which can increase the time required, or might introduce a rival NPC trying to foil the plans).
The Prodigy Feat from XGtE enables you to learn one language, one skill proficiency, one tool proficiency and expertise in a skill you are proficienct in.
The Linguist Feat from the PHB gets +1 to Intelligence (to a max of 20), 3 additional languages, the ability to write in ciphers.
If we assume each benefit in these feats is equal to one language, that gives us a ballpark estimate of 4 languages to learn a feat.
So that gives us a minimum time of 40 workweeks - Max(Intelligence Modifier,0), before complications. It also has a 4 x 10% chances of getting a complication (roll on 2d10 4 times and if any of them are 10 or less a complication happens in that set of 10 workweeks).
If it's not given as a reward this will cost the character in the region of 875gp - 1,000gp.
A story reason for giving the ASI as a reward from a supernatural being would be appropriate here. This should likely have arisen from a quest that the party has completed for this being, which should probably be as a result of several Hard/Deadly encounters.
In Tier 2 the character will get an ASI at level 8...they can use it to bump up their stat/gain a feat.
Options currently available in Core Handbooks (PHB, DMG, XGtE, MM) as of this post
The DM has a wide range of options for Permanent increases, a decent number of which are Magic Items.
Class Levels [PHB]
When a character reaches specific levels in a class they gain an Ability Score Increase (ASI). This gives the character the ability to increase one score by +2 or two scores to +1 up to a maximum of 20.
It is important to note that the class levels are distinct to character level. A character is limited to 20 levels in total. So if a character multiclasses they may miss out on one or more ASIs that would have been gained from increased class levels.
Additionally specific classes include a capstone ability (benefit for stying in a class for the full 20 levels) which offer significant ASI benefits (namely the Barbarian's Primal Champion Ability).
Wish is the only spell that might allow a permanent increase, and only if the DM allows it, and the caster will need to roll for stress to see if they can ever cast wish again.
Magic Items [DMG]
Minor Items - Rare
- Bag of Beans (one of the effects potentially enables a user to increase one of it's lowest ability scores by 1)
Major Items - Very Rare
- Manual of Bodily Health (+2 Constitution)
- Manual of Gainful Exercise (+2 Strength)
- Manual of Quickness of Action (+2 Dexterity)
- Tome of Clear Thought (+2 Intelligence)
- Tome of Leadership and Influence (+2 Charisma)
- Tome of Understanding (+2 Wisdom)
Major Items - Legendary
- Deck of Many Things (Star card: permanently increase a score by 2, up to a max of 24)
Feats [PHB] & [XGtE]
If your DM allows you to gain feats through downtime training, a number of these grant an ASI. (PHB: Athlete, Actor, Durable, Heavily Armored, Heavy Armor Master, Keen Mind, Lightly Armored, Linguist, Moderately Armored, Observant, Resilient, Tavern Brawler, Weapon Master
XGtE: Dragon Fear, Dragon Hide, Dwarven Fortitude, Elven Accuracy, Fade Away, Fey Teleportation, Flames of Phlegethos, Infernal Constitution, Orcish Fury, Second Chance, Squat Nimbleness)
Deities (DMs) can bestow blessings on characters, as an alternative to loot drop after a battle, which can include ability score increases (the example ones in the DMG suggest +2 with up to a max of 22 is appropriate for a blessing). Characters can only benefit from one of each type of Blessing at once (eg Blessing of Health: +2 Constitution up to a max of 22). There are other Blessings given as examples in the DMG.
Artifacts can increase an ability score by +2 (up to a max of 24) - This may or may not be temporary depending on the artifact.
eg Book of Exalted Deeds
Epic Boons [DMG]
These can be awarded to 20th level players by the DM. The method for awarding them will vary by DM, but one suggested approach in the DMG is to award one for every 30,000XP above 355,000XP.
An alternative the DMG suggests that the DM can award an ASI (+2 to one stat or +1 to two stats, up to a max of 30), instead of an Epic Boon.
Temporary increases are contrained to Magic Items/Potions. Artifacts can also give temporary increases while attuned.
Various spells, like Animal Shapes, Polymorph, Shapechange and True Polymorph replace character stats with the target creature stats.
Magic Items [DMG]
These magic items enable the character to increase their ability score while using the magic item, some/all of which will require attunement (I've not checked if they do).
Major Items - Uncommon
- Gauntlets of Ogre Power (19 Strength)
- Headband of Intellect (19 Intelligence)
- Potion of Hill Giant Strength (set Strength Score to 21)
Major Items - Rare
- Amulet of Health (19 Con)
- Belt of Dwarvenkind (+2 Con to max of 20)
- Belt of Hill Giant Strength (Set strength to 21)
- Potion of Frost/Stone Giant Strength (set Strength Score to 23)
- Potion of Fire Giant Strength (set Strength Score to 25)
Major Items - Very Rare
- Ioun Stone: Agility (+2 Dexterity, Max 20)
- Ioun Stone: Fortitude (+2 Constitution, Max 20)
- Ioun Stone: Insight (+2 Wisdom, Max 20)
- Ioun Stone: Intellect (+2 Intelligence, Max 20)
- Ioun Stone: Leadership (+2 Charisma, Max 20)
- Ioun Stone: Strength (+2 Strength, Max 20)
- Belt of Frost/Stone Giant Strength (Set strength to 23)
- Belt of Fire Giant Strength (Set strength to 25)
- Potion of Cloud Giant Strength (set Strength Score to 27)
Major Items - Legendary
- Belt of Cloud Giant Strength (Set strength to 27)
- Belt of Storm Giant Strength (Set strength to 29)
- Hammer of Thunderbolts (+4 Strength while attuned, up to a max of 30)
- Potion of Storm Giant Strength (set Strength Score to 29)
Class Abilities [PHB/XGtE]
Wildshape replaces the physical stats of the Druid with the relevant Beast stats while in Wildshape.
Conditions as a result of interation with specific monsters [MM]
Both lycanthropy and vampirism can be applied as a template to a player character, with DM discretion. Both will potentially alter the ability scores of a character.
For example vampirism increases the Strength, Consitution and Dexterity scores to 18 if they weren't already higher.
Both can potentially be removed through various actions in an adventuring career.
Artifacts can increase an ability score - This may or may not be temporary depending on the artifact
eg Hand of Vecna (while attuned Strength Score is set to 20)
[ASI]: Ability Score Increase
[PHB]: Players Handbook
[XGtE]: Xanathar's Guide to Everything
[DMG]: Dungeon Master's Guide
[MM]: Monster Manual