The Famous Houserule of Taking Average HP
I did not come up with this, but since it would be good to have it onsite to link to, here is the method for taking average HP when your character levels up:
When a PC gains a level, they can (with their GM's blessing) opt to just take the average amount of HP instead of rolling the newly acquired Hit Die. The average can be found by taking the die's minimum result, adding it to its maximum result, and then dividing it in half. If fractional HP is created, then it is held in reserve until a full point is obtained.
Let's use a Fighter as an example. A Fighter uses a d10 Hit Die. When created at Level 1, the Fighter got to take the full 10 HP, thanks to the rule that PCs always get max HP at 1st level. Upon reaching Level 2, the Fighter's player opts to take the average instead of risking a roll. On a d10, the lowest one could roll is a 1, and the highest is a 10. Together, that makes a total of 11, which when divided in half is 5.5.
So the Fighter gains 5 HP (plus any modifiers), and that .5 HP is held in reserve until it grows into at least one full Hit Point. When the Fighter reaches Level 3 and the player chooses to take the average again, the Fighter once more gains 5.5 Hit Points, but when that is combined with the other .5 that is already waiting in the wings, it just becomes an even 6 Hit Points.
If the Fighter doesn't multiclass, and the player keeps choosing to take the average, then this pattern will just continue, with the Fighter earning an alternating base of 5, 6, 5, 6, 5, 6, etc. each level. And that's how you take average HP.
Note: There are some versions of this method out there where the HD that you are reaching is what determines whether or not you take a fractional Hit Point. According to this way of doing it, you always take a rounded down version on an even HD, and a rounded up version on an odd HD. The reason that this way of doing it is not correct is that multiclass characters gain an edge here where it is possible for them to slightly game the system by carefully choosing the order in which they take their levels in different classes.