Let's look at some possible examples, building on @ben's answer and @ Knudsen's answer.
We know that the MarCo cubesats were able to navigate from Earth to Mars, with
- attitude control via reaction wheels and cold gas thrusters
- science data and image collection
- communication directly with Earth via a unique pop-up flat high gain antenna
- 70W of solar power at 1 AU via two deployable solar panels plus battery storage
- standard 6U form factor
for more see this answer and links therein.
So let's adopt the MarCo design. They didn't provide their own propulsion, so let's add a propulsion system directly to MarCo's 6U, 14kg initial configuration, and call it 10U and 22 kg. The extra 4U volume is mostly for engines and extra propellant, the extra 8 kg mass budget is for engines and additional solar panels for more electric power, especially out near Mars and a whole bunch more propellant!
Looking for at least apparently existing cubesat electric propulsion systems that you could put in a 3U cubesat today (or soon), the first one that came up in my search is the IFM Nano Thruster for CubeSats. I am sure thee are other options out there, let's just use this as an example. According to that page:
Dynamic thrust range 10 μN to 0.5 mN
Nominal thrust 350 μN
Specific impulse 2,000 to 5000 s
Propellant mass 250 g
Total impulse more than 5,000 Ns
Power at nominal thrust 35 W incl. neutralizer
Our cubesat will have enough electric power for two engines at 1 AU, since we've expanded the form factor by 4 U and mass budget by 8 kg will allow for larger solar panels.
Our two off-the-shelf engines with 250 g propellant tanks each can provide a total impulse of as much as 10,000 Newton seconds. With an average mass of about 20 kg, that only provides a delta-v of 500 m/s. But how much do we need?
Luckily there's an existing mission that addresses this already! Answers to Going from LEO to lunar using only low-thrust ion propulsion - can it be done? say that the SMART-1 mission has done this already!
According to that article the propulsion system used to provide a trajectory from GTO to the Moon (crash landing) demonstrated a total delta-v of about 3,900 m/s.
Luckily we'd added 8kg to our mass budget, so if we'd added an extra 5 kg of propellant we'd have a total impulse of 100,000 Newton seconds and a delta-v of about 5,000 m/s.
A back-of-the-envelope calculation starting with a MarCo-like cubesat with demonstrated capability of going from Earth all the way to Mars, augmented from 6U 14 kg to 10U 22 kg with two existing engine designs and another 5 kg of propellant, we can get from GTO to the Moon using solar-electric propulsion.
The extra delta-v allows for maneuvering near the Moon and doing a bit of sight-seeing and selfie-taking.
Alternatively you could use the extra delta-v to boost yourself from LEO to GTO, allowing for a more standard cubesat deployment option as long as the inclination were not too high. That would probably need another few kg of propellant, so it's marginal. Best way to proceed would be to piggy-back on one of the many existing launches to GTO in a similar way to how the MarCo's piggy-backed to the transfer orbit to Mars.
Source: MarCO: Mars Cube One
below: Source: Emily Lakdawalla's Planetary Society blogpost MarCO: CubeSats to Mars!
Found in this answer.
MARCO SPACECRAFT: Engineer Joel Steinkraus stands with both of the Mars Cube One (MarCO) spacecraft at NASA's Jet Propulsion Laboratory. The one on the left is folded up the way it will be stowed on its rocket; the one on the right has its solar panels fully deployed, along with its high-gain antenna on top.
An alternative, future propulsion system with even higher Isp and therefore needing less propellant mass:
An encouraging video: