I have not tissue cultured any plant. Researchers have published their methods specifically for daylilies in a number of different journals. There are probably professional laboratories that will tissue culture a plant for a fee. There are probably some that specialize in daylilies. I personally do not know of any.
To get increase from a plant it needs to receive optimum growing conditions and to use all the energy it receives to grow rather than to flower or produce seeds. If a gardener/hybridizer wants to push a plant to increase then the plant should not be allowed to flower. Remove the scape as soon as you see it. Usually that also means pushing the growth with high nitrogen fertilizer or just nitrogen fertilizer throughout the growing season. Keeping the plant mulched to conserve soil moisture is also a good strategy.
Tissue culturing a plant pretty much requires a laboratory setup and dedication to chemistry lab techniques. There are web-sites that encourage "kitchen table"/home tissue culturing and a dedicated amateur can manage to tissue culture plants but it is not simple. It probably has a steep learning curve with many losses along the way.
There is a method which involves cutting a crown into quarters or smaller sizes and then using a plant hormone such as BAP to get increase. However, simply cutting a crown into quarters should produce four new plants (even without using any plant hormones) if the pieces are treated optimally. This is like cutting the top off a stem of any plant, say a rose. What happens next is that the stem can no longer grow until one or more of the buds further down sprouts. The same idea applies to daylily crowns. In the centre of the leaves of a fan is the growing point. If that is destroyed then the crown can produce new buds from in-between where the leaves were. What one ends up with, if one is lucky and say all four pieces produce new buds and all four survive, is four new fans where before there was only one. The catch is that the four new fans are usually smaller than 1/4 of the old original fan.
