On meaning: ΕΥΦΡΟΣΥΝΟΣ doesn't mean 'be cheerful' or 'enjoy your life', it means 'cheer-bringing' or 'cheerful', and implies someone or something with masculine grammatical gender.

I advise you that getting a tattoo in a language you don't understand is very high-risk. It isn't just about meaning or spelling: it's about context, design, and layout, and neither you nor the tattoo artist will have any idea what variations are and aren't sensible. Some specific examples:

• The word is gendered. Is it the correct gender for you? Does it matter? I can't advise you on that.
• The letter forms in your question are in a modern type face; the mosaic uses ancient letter forms. Which to use? Is it OK to mix and match?
• In your post ΕΥΦΡΟΣΥΝΟΣ is written in a single line; that isn't how it appears in the mosaic. Is either OK?
• What about lower case letters, variant letter forms, typeface, diacritics? Will you mind if some letters appear in a different typeface by mistake? Because that happens pretty regularly on computers.
• Because Roman letters are so familiar to you, neither you nor the tattoo artist is likely to notice any accidental substitutions.
• Why would you trust me about the word's meaning? I may have an /r/AskHistorians flair, but that doesn't mean I never make mistakes.

I accept that some people are willing to take the risk anyway: I'm advising you that all of the above risks are very real. There are plenty of tattoo designs out there, even done by people who supposedly specialise in Greek designs, that run foul of stuff like this. Here are some illustrations from a couple of webpages that recommend Greek tattoo designs: roughly half of them do what the artist wanted.

• This designer doesn't know the difference between different diacritical marks in Greek, and so uses a breathing mark for accents and breathings indiscriminately (even though breathings aren't required in modern Greek).
• This one is probably meant to mean 'wanderer', but it really means 'male traveller'. On a woman's arm.
• This one means 'be manliness', which I doubt is what they were going for.
• Complete gibberish (from this page). It looks like the result of really bad automated OCR.
• Also gibberish. This one has taken a meaningless placeholder text used by graphic designers and just transliterated it into the Greek alphabet for no apparent reason.
• These four friends all got the same tattoo, intended to mean 'friendship'. All four have the same grammatical error.
• This one is real Greek, and comes from a tagline for a film ... but it's originally a quotation from a Latin poem.
• There's an entire subreddit devoted to people misusing the Greek alphabet, and there are a couple of horrific tattoos there: this one, and this one.

Obviously I can't stop you from getting the tattoo, but also obviously I can't be there to answer the tattoo artist's questions for you, or to make sure that the design you present to the tattoo artist has been reproduced accurately.