Play Without Map and Minis - Theater of the Mind
Some tables play without a grid. In 5e, unlike some other editions, the grid is an optional rule, not a core rule. It is so common that we often take for it for granted that people are using grid squares. 5e can be played entirely "theatre of the mind" without maps and minis at all. I personally have only played theatre of the mind with other systems; but there is nothing special about minis and map.
Play Without A Grid - Gridless
Some tables play instead a rulers (sometimes templates) and math, and you really can move any distance less than your total movement. I've see some terrain crafters on YouTube that talk about the freedom they feel going "gridless". This would have you move 4/5"ish on the map, which isn't a big deal at all.
With a Grid
There are four ways I've seen at people use the grid for odd increments of movement:
If you can't move enough to leave the square, you effectively didn't move. Any movement under 5ft just "doesn't count" and is just shifting around in your square. I believe this is what a strict reading of the optional rules would have you do as Exempt-Medic's answer explains. This would have you not move at all.
Center "of Mass"
If you control a space 5ft, the midpoint is where you "are" so if you move more than 2.5ft, you're effectively in the next square and it is effectively rounding to the movement to the nearest 5ft. This seems the most "fair" but I don't think it is as common as the previous one. This is also similar to what the optional rules say about when damage from an area effect should apply, that if the majority of the square is covered by the effect a creature in the square is affected, but if less than half the square is covered, a creature in the square is not affected. This would have you move up one square.
I don't think I've ever seen this played for movement, but I have seen people play it for area effects because they like "bigger fireballs" and what not. I don't think this is a likely answer for movement less than 5ft. This would also have you move up one square.
Divide the Grid
You can live with a character not quite in the grid system. This would have you move over 4/5th a square, or I've seen some just handwave "It'll takes two turns movement to leave get there, using your full movement each turn" because they don't the mini not clearly in a grid box.
Whatever You Do, Keep it Consistent
How a DM chooses to handle it, they should be consistent about it when it comes up. What you do for a PC, you do for monsters and vice versa. They can change how they handle the uneven movement amounts if they choose, but it should be communicated to table and from then on they should do it the new way for any of the rate times it comes up. That said, it is a rare thing and many DMs won't run into it, because most of the rules are designed around numbers divisible by 5 specifically to be grid-friendly.