“Who remembers their birthday?”
“Who remembers one thing another person in this room did that make you
Neither of these sound awkward*. The choice between third person and second person when addressing a crowd is a matter of what emotional effect you wish to convey, not a matter of correct or incorrect grammar (except to those rather old-fashioned purists who object to "they" being used as a gender-neutral third person singular pronoun). To me, "you" sounds more personal, as if you are reaching out to each person in the audience as an individual, whereas "they" has a less emotional tone - which may be more appropriate in some circumstances.
The thing wrong with your sentence “Who remembers one thing another person in this room did that make they smile?” is not the use of the third person plural in itself but the fact that you used the third person plural subject pronoun "they" when you should have used the object pronoun "them".
There's an explanation of subject and object pronouns here.
*Actually, one of them does sound awkward, as pointed out by Edwin Ashworth in the comment below. I failed to spot that "make you smile" in the second sentence should be "made you smile".
That point aside, the important thing is that both “Who remembers one thing another person in this room did that made you smile?” and “Who remembers one thing another person in this room did that made them smile?” sound quite natural for a speech.