Get more cultural tradition. It works with Great men the same way Army/Naval tradition works with Army/Naval leaders.
Though some great men have hybrid tradition requirements. Army Tech requires Culture + Army tradition, Naval Tech req. Culture + Naval Tradition.
There are a few ways to increase your Cultural Tradition(CT).
Buildings: Fine Arts Acads,+1%, really only viable later game. Glorious Monument +1%, limited to 1 in your country, again viable later on.
Magistrates:This will be the bulk of your CT early on. Under your culture tab theres certain decisions you can enact, Tapestry, Renaissance ones, Endorse Culture, Epic, etc. But the majority of your early magistrates will(and should imo) be spent on Paintings. Not to discount the other options, but they usually have a set duration and can't be selected again after thats up(if at all). This is +5 CT per painting. This is a very useful thing to know early game when you don't have the money/tech to build buildings and thus your magistrates are essentially taking up space. Make them work for you!
Ideas
atron of the Arts will give 3% CT. This offsets the decay of CT which is also 3%. How ever there are generally more important Ideas to take(Drill, Trading, Colonists, QFTNW, etc) given the next option.
CAD/GI Switch:When I reach Govt. tech 4 I always do the fallowing. I Choose Church Attendance Duty, reduction in Stability costs. In itself the idea can be powerful for large multi-ethnic and multi-religious empires, of which few countries fit the bill at the point you reach govt. 4, what makes this important is not the stability but the Gilded Iconography idea that is enabled under your cultural tab. CAD+1 Temple(anywhere in your country iirc) enables this decision which does a number of things(Prestige, missionary chance, etc), but most importantly gives you 3% Cultural Tradition. The same as the PotA Idea and the same as the decay. Thus you can offset you decay and make your spending magistrates on Paintings and other cultural decisions more efficient. After you get GI you can switch CAD, or not, if the situation warrants it(more often then not it does) for a better Idea, trading or Drill for example. The Gilded Iconography will stick around after you switch and give you the benefit of essentially a free National Idea(and then some).
Utilizing the CAD/GI switch and proper use of magistrates I run a constant cabinet of 5 to 6 star advisers.