I have a character with Thieves Tools and it includes a small mirror and my DM is putting my 4 player 5th level character group of players against it and I'm planning on reflecting the eye ray back at it. Would this work?

RAW: nope.

RAF: up to DM.

Personal: nope.

EDIT to expand on my personal choice to now allow if I was the DM: I would not allow it because you are effectively turning a CR 13 encounter into a CR 0. The idea of just making yourself immune to the eye rays and being able to use them against the Beholder, just by having a mirror, is insanely OP and makes the whole encounter utterly pointless. But different DMs may rule differently. 

Generally in 5E things only do what they say they do. If you could reflect a beholder's eye beams with a mirror or other reflective surface, it would say so. 

Your DM may or may not allow it, but RAW this wouldn't work and i wouldn't let it work in my games either. 

The various "Ray of X" spells can't be reflected either. Plus, this would presuppose being able to 'catch' the ray with the mirror (which should at the very least require a check, and arguably use up your reaction) and the mirror being large enough that the ray can be caught fully. The mirror thing is a neat idea that would work well with monsters or challenges specifically designed for it (the medusa of Greek mythology comes to mind, obviously) but I wouldn't tack it on on an existing monster without some major adjustments that'd make it specific to the encounter rather than a generally applicable option.

Aside from the simple answer of "no way" as already explained, it sounds like you might not be clear on the type of mirror in a thieves' tools kit. It's "a small mirror mounted on a metal handle" intended for stealthily looking around corners and would at most be the size of a typical car's side mirror. It has to be easily stowed in a small pack that can be easily carried by a person moving stealthily while climbing, crawling, and tumbling about as well as being unobtrusive enough to not announce "I am carrying tools for the purpose of breaking and entering places I am not supposed to be" too anybody that looks at you walking down the street. It could even be implied to be even tinier, along the lines of the mirror a dentist uses to look at the back of your teeth, in this case used for looking under doors (or just being very hard to see poking out around a corner or such. This is not Perseus' shield that he used to make Medusa look at herself with her petrifying gaze (which was never described as anything resembling a "ray" in the original myth). 

I give you points for creative thinking, but just no, for a lot of reasons. As stated by others already, you're free to suggest it but don't be surprised if your DM says it ain't going to happen.

Even if this did work, imagine how difficult it would be. You'd need to place the small mirror directly in the path of the ray without letting it hit any part of you, then angle the ray back perfectly so it hits the beholder. So for it to totally work you'd need at a minimum:

• a successful saving throw to avoid getting hit
• high DC sleight of hand or something similar
• failed save by the beholder