Euodoo means to have a good road (eu + hodos), i.e., to prosper, to succeed, to have things go well. It can describe a good journey but is used figuratively for prosperity and success in all endeavors. Paul uses it in 1 Corinthians 16:2 for the financial prosperity of believers, and in Romans 1:10 for his hoped-for journey to Rome.
The most famous use in some translations is 3 John 2: 'I pray that you may prosper [euodoo] in all things and be in health, just as your soul prospers.' This has been misused to teach a 'prosperity gospel,' but in context, John is using a standard Greek letter greeting — a polite wish for good health and safe travels. The theological insight is different: John wishes bodily prosperity to match Gaius's already-evident spiritual prosperity. The norm he assumes is that spiritual health (soul prosperity) is the primary reality; physical health is secondary. God's deepest prosperity for us is always a flourishing soul.