HP Gained at Level Up

Rules As Written

When you gain a level, you increase your maximum hit points by rolling the hit die granted by the class you are leveling up and adding your constitution modifier (PHB 15).

House Rule

If you roll less than half on the above roll, you get half HP instead; half is the lowest you can roll. For example, if your hit die is a d8 and you rolled a four or lower, you would treat that roll as if it were a 4.

Rationalization/Reason for the Rule

Having more HP than average is an advantage, but not as big a one as you may expect. Having less HP than average, on the other hand, can cripple a character to the point where they're unplayable. I'd rather have slightly overpowered characters on average than characters that are crippled & useless, since I can just up the difficulty of encounters slightly to compensate for the former.