I have a character who for all intents and purposes, looks sleepy and tired. She has bags under the eyes, swaying to her walk, slow speech and reaction time, etc. In reality, she has the current stat array of

Str: 13

Dex: 16

Con:15

Int: 19

Wis:17

Cha:10

So she just has the appearance of being sleepy. My question is, what kinds of bonuses or downsides would she have in social situtions (ignoring stats) Like is it easier to find an inn by talking to NPCs, does she need extra time to think etc. My DM is at a standstill in this because he likes my character and wants to add a mechanic for this kind of interaction, but doesn't want anything too powerful. Think possible Level 1 homebrew class feature if possible.

Thanks in advance!

Basically give them permanent 1 level of Exhaustion as this specifically represents a character who is very tired.

Or if this is not genuine tiredness and they are just being lazy (there's a difference) consider the same effect of disadvantage on ability checks but not actually be a level of exhaustion so you are making the choice to have this disadvantage out of laziness and can make proper skill checks when really trying (for situations that would prompt you out of laziness like your close friend about to die and needing you to save them).

Personally I wouldn't be that concerned with actual mechanics to hinder what the character can achieve (unless you really want that) but use it more as a narrative tool in how to show the way she executes things.

Maybe she shows a kind of disinterest/distraction due to being 'tired', while it doesn't effect rolls per say, but NPCs may pick up on it and give more specific or to the point answers or fewer details when responding.  Perhaps comments are often made, or she gets directed to the nearest Inn in quite often as a side bar to the conversation.
Typically I like using character quirks (like the one mentioned) when explaining how poor roles failed (or didn't really succeed as well as normal) is a great way to give failure that nice bit of colour.  Maybe she didn't get the information as she dozed off in the middle of things, annoyed the NPC by yawning often, or many other ways to give the 'why' a roll result was poor.

An alternative is to have a secondary die to use to simulate disadvantage, but as an optional result to choose on occasion.  Essentially you have say a Red and Grey d20, normal rolls use the red, but the grey one is the "sleepy die".  Anytime the grey die is lower than the red you could chose to use it for the narrative, you wouldn't ever have to take the grey die like other rolls that give disadvantage but it is used more for the narrative moments you think could impinge on the situation.

As to semi-mechanical quirks, look at the background flavour abilities, they generally offer some nice character features but don't have much extra power.  The DM and yourself could even come up with a custom background to cover the concept as well if it would fit.