As both a player and DM, I love Hero Points. In the current group I'm DMing, we only had 3 players (up to 4 + DM now) and everyone was somewhat inexperienced so HPs gave them additional survivability at low levels.
As mentioned, they give a bit of safety net if the dice are really going against the players in a given session. I also like that they give me - the DM - another resource I can chip away at in addition to hit points. As they use up their HPs, the tension increases since they know they won't have that crucial HP to save their butts. I'm also fine if they save them for boss battles. In my experience, the addition of HPs makes boss battles that much more epic. Finally, I know that if I throw something at the players that might be a bit too challenging than I wanted - or an encounter turns out to be unexpectedly difficult - I know it won't turn into a trainwreck.
I started with the rules in the DMG for HPs but I modified the rules for my game, borrowing heavily from 3ed Trailblazer:
Hero Points
All characters have 5 Hero Points. Hero Points are refreshed when a character gains a level; unused points are lost.
Spending Action Points
• Improve an attack roll, ability check or saving throw. Roll a d6 and add the result to your d20 roll. You may only use a Hero Point to improve the result of a roll before the DM informs you of the outcome of the roll. You may only use one Hero Point per roll to improve any given d20 roll.
• Negate a critical hit scored against you by an opponent.
• Improve your AC. As a bonus action, spend a Hero Point and roll a d6. The result is added to your AC until the start of your next turn.
• Heal in combat. As an action, spend a Hero Point and spend up to half your Hit Dice to heal.
• Whenever you fail a death saving throw, you may spend a Hero Point to convert the failure into a success. If you roll a 1, you can only convert one of the failures.
Party Pool
A Party Pool is a pool of Hero Points shared by the party. The Party Pool represents the characters learning to work together as a team.
Hero Points in the Party Pool function normally and may be used by anyone in the group. When Hero Points refresh (upon gaining a level), each player may contribute 1 Hero Point from their own total to the Party Pool.
There is no maximum to the Party Pool, and Hero Points in the Party Pool are not lost when Hero Points refresh. However, if the current number of Hero Points in the Party Pool is equal to or greater than the number of characters in the party, a character can not contribute to the Party Pool. This may result in not all characters being able to contribute a Hero Point to the Party Pool.
The DM may also choose to award Hero Points to the Party Pool when the party performs heroic deeds or accomplishes difficult quests, especially those that are optional.