Option 1, Stick to the Rules
This means no rerolls at all. Most editions of Dungeons of Dragons don't even mention the topic of rerolls. (Several editions also don't mention Critical Hits, but I will get back to that topic later.)
Option 2, Allow Rerolls as a House Rule
This means you need to come up with a rule system to govern how often rerolls can be used and when. Too many rerolls means nothing bad ever happens to the PCs and they use their rerolls willy nilly, too few is arguably better because it means players will only use those rerolls when they really need them in order to survive, and thus will conserve their rerolls for when something dangerous happens and they absolutely need to use a reroll.
Thus as a DM if you decide to allow rerolls you need to come up with a system for using them and how many rerolls are available.
1. Once Per Level
One reroll available, once per character level. This assures the character will have very few rerolls and will attempt to conserve their one and only reroll until they manage to level up to next level and only use it in do-or-die situations.
2. Flat Number at Level One
The character starts with 5 rerolls (for example), which they can use whenever they want, but when they run out they no longer have any more. This increases their survival chances at low level but as they get to higher levels they start to run low on rerolls and have to conserve them.
3. Rerolls based on Charisma Score
12 Charisma equals 1 reroll per session, 14 Charisma equals 2 rerolls per session, 16 Charisma equals 3 rerolls per session, etc. This system gives a more plentiful number of rerolls, but also ensures that Charisma is no longer a dump stat for low stat rolls. Suddenly a high Charisma is a very valuable stat and players are encouraged to use it more often and perhaps realize the potential of playing a high Charisma character.
4. Add a Luck Score
Similar to Charisma above, characters roll a Luck score during character creation and this governs how Lucky their character is. This is similar to how some games might also use ability scores for Honour, Sanity, Speed, Comeliness/Attractiveness, Social Rank, etc. eg. In 1st edition AD&D many players use Comeliness as a representation of a character's attractiveness, a separate ability from Charisma. In 5th Edition D&D Sanity and Honour are listed in the DMG as possible optional ability scores. In other games like Palladium Speed is listed as an ability score, so this is not a new concept.
5. Reroll Cards for Good Roleplaying
Make up a set of "One Free Reroll" cards and print them on cardstock. Eight or ten is a good number of cards to make, depending on the size of your group. Then give them out to players sparingly whenever they do a good job roleplaying. Don't award them for killing monsters. Only reward them when they do something noteworthy and memorable roleplaying wise. This encourages players to roleplay their characters, stick to their alignment, and also rewards smart thinking within the bounds of their character. It can also be used to reward good story development, avoiding the temptations of metagaming (especially when it is really tempting), and "doing the right thing" within their alignment. Chaotic characters being able to do almost anything, this really tends to reward characters who are lawful or neutral and behaves appropriately, but again I should remind you to only hand out these cards sparingly.
Also note that if you run out of cards (eg. if you give all 8 or 10 of them to the players) don't give out any more. The players cannot hoard the cards if they know there is a small supply of them, and also recommend capping individual players so they cannot have more than 2 reroll cards at any one time.
As a sub-rule, trading reroll cards should also be prohibited. As is giving them away for favours.
6. Luck as the result of Piety
Under this house rule, Luck can only be granted by the gods. Donating wealth or magical items to the church (it doesn't matter which church and it also doesn't matter which class the character is) can earn the PC a reroll. This encourages every character to choose a deity to worship, even if they are a lowly thief. The size or kind of donation given doesn't always matter either. The character could be donating gold, donating magical items, or even donating their time by volunteering to go on quests for the church or perform services on behalf of the church. If the gods are pleased with the results, they award a Luck Reroll, which could be recorded either as Luck Points or it could use "One Free Reroll" cards like #5 above.
7. Combinations of the Above Rules
You might even decide to employ more than one of the House Rules above and use them together.
eg. 1 and 2 work well together, as the rerolls are limited and eventually run out - and yet each character gets a new reroll every time they level up, which they will hopefully conserve for when they need it.
3+5 or 4+5 also work well in combination. I currently use 3+5 in my current campaign as it benefits both good roleplaying, which I like to encourage, and it encourages players to not use Charisma as a dump stat. I also use House Rules for Charisma to add benefits during large scale battles with followers, thus Charisma is exceptionally valuable if PCs choose to use their Charisma score to its full potential.
5+6 also makes an interesting combination, as it could be used with the same "One Free Reroll" cards, and thus fits seemlessly. Again, keep the number of reroll cards limited and don't allow players to hoard them in numbers more than 2.
BONUS, FIVE ALTERNATIVES TO CRITICAL HITS
1. Instead of double damage for rolling a natural 20, make it triple damage. The potential for way more damage makes everything potentially more dangerous.
2. Double damage on a natural 19, triple damage on a natural 20. Makes critical hits more common, but also ups the damage dramatically.
3. When a natural 20 is rolled give the PC a choice: Trip, Stun, Disarm, Push or Double Damage. A trip causes the enemy to fall to the ground. Stun makes them dazed and unable to attack for one round, but they can still defend themselves. Disarm causes their weapon or shield to be knocked away from them. Push moves them 1 square in any direction the PC wishes. And Double Damage is the normal effect we are accustomed to.
4. Give them a bonus attack. If they succeed on the 2nd attack, roll damage accordingly. Note that this means that if they roll a 2nd natural 20 they would get a 3rd attack and possibly a 4th.
5. Roll Percentile Dice for an Auto-Kill. The base chance for an Auto-Kill is 50%, but it is modified by the character's level versus the target's Hit Dice. If the attacker is level 5 and the target has 10 HD, the chance is only 25% because they are 5 levels lower than the target's HD. However if the target is 2 HD and the attacker is level 6, then they have a 70% chance of killing their target because they are 4 levels over the target's HD.
6. Random Bonus Damage. Roll a d6, if you roll a 1 or 2 then the attack does double damage. If you roll a 3 or 4 then the attack does triple damage. If you roll a 5 then roll quadruple damage. If you roll a 6 then roll both quadruple damage and check to see if it is an Auto-Kill as per #5 above.
Note: Making critical hits that much more dangerous makes for very cautious PCs. They suddenly realize that even a dagger can do a lot of damage in the hands of an enemy if they manage to score a natural 20 and suddenly do 3d4 or 4d4 or Auto-Kill instead of their normal 1d4. Suddenly having rerolls sounds like a good idea, but unfortunately rerolls cannot be used to reroll an enemy's attack.
Happy playing!
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