Old 1E player now getting back into 5E. What are HIT DICE? I understand hit points and a 3rd level player has rolled 3 times for HP ( I do like the MAX at first level and then roll option) but what are Hit dice and why do they matter?
Generally, template describes the situation if there will be any Hit Dice size change . Wizards online Types and Templates defines this as follows (along with changeing Constitution score)
Before applying any Constitution changes, check the Hit Dice and Hit Points entry in the template. Some templates change the number or size of the creature's Hit Dice or both. Some templates change previously acquired Hit Dice and continue to change any additional Hit Dice the creature gains. Most templates, however, change only the creature's racial Hit Dice (that is, the Hit Dice it has before adding any class levels). Most templates are fairly explicit about what happens to the creature's Hit Dice, so just follow the instructions in the template.
When you know the size and number of the creature's Hit Dice, recalculate the creature's hit points using the modifier from the creature's new Constitution modifier for each Hit Die (whatever its size).
In your example, you re-roll hit dice with new hit dice size.
From D&D 3.5 Monster Manual Hit Dice entry description:
This line gives the creature’s number and type of Hit Dice (the die rolled to generate hit points), and lists any bonus hit points. A parenthetical note gives the average hit points for a creature of the indicated number of Hit Dice.
D&D generally defines base HD and additional HP (CON bonus and other bonuses if available) for creatures, but also offers a pre-calculated value for HP of that creature. So you can either calculate it or use pre-calculated value.
That means, game mechanics leave the final decision to DM (in most cases), and it is up to him. (Such as; in my games, I generally offer players 3/4 of the die they roll for their HP when they level up, and calculate NPC HDs in a similar way.)
Better you discuss that with your DM and all of your players and find a way that suits everybody.
It's funny how things like this go crazy when you look at how different editions on different languages work.
Hit Dice, in Portuguese, was translated (incorrectly) as "Dado de Vida", or "Life Dice". In portuguese, you don't have "Hit Points", you have "Pontos de Vida", or "Life Points".
So, at least for portuguese-speaking people, associating "Life Points" with "Healing"(Gaining "more life") is trivial.
The same don't happen on english, however. I had a player who was on the same situation a few days ago, having this same issue (albeit on another system). He simply couldn't plug on his mind that the same value for "Hit Dice" would be used for Healing.
So, we went up to try to find a few mnemonic ways to remember that easily. We came up with something... fuzzy. We went to write some verses, to find something easy to remember. We came up with this (Keep in mind that I'm a GM, not a Poet, so yeah, that's a heck of a silly thing):
When Ones roll plenty, and the battle goes bad,
No fear I have, I have to say that,
One of them is a Cleric, and good one in fact,
Always prompt, ready to act
And because of him, I always remember
That to my well being, on that dreaded chamber
Dear to the Heart, My hit dice should I Keep
'Cause with them he will heal me, in time of need
(It's actually bigger than that, but I removed the non-relevant parts)
The last two verses did the trick, when you read them in a "Dr. Seuss"-like way.
Yeah, it's bad, but hey, it worked!