I would follow the same process to choose a twin mattress for a daybed as I would for choosing any other twin mattress except I would add a few things to your testing.
The first place I would start your research is the mattress shopping tutorial here which includes all the basic information, steps, and guidelines that can help you make the best possible choice … and know how and why to avoid the worst ones.
Two of the most important links in the tutorial that I would especially make sure you’ve read are post #2 here which has more about the different ways to choose a suitable mattress (either locally or online) that is the best “match” for you in terms of “comfort” and PPP (Posture and alignment, Pressure relief, and Personal preferences) that can help you assess and minimize the risks of making a choice that doesn’t turn out as well as you hoped for and post #13 here which has more about the most important parts of the “value” of a mattress purchase which can help you make more meaningful quality/value comparisons between mattresses in terms of suitability (how well you will sleep), durability (how long you will sleep well), and the overall value of a mattress compared to your other finalists (based on all the parts of your personal value equation that are most important to you).
When you are sitting on a mattress your weight will be much more concentrated than when you are lying on a mattress and you will tend to sink in more deeply when you are sitting so if you are using a mattress as a daybed where you will be sitting more frequently then I would also make sure you add an additional “S” to your regular testing for PPP (Posture and alignment, Pressure relief, and Personal preferences) because you will also need to test the mattress for how well it works for you when you are sitting in addition to testing the mattress for sleeping in all your sleeping positions. In other words you will be testing for PPP + “S” (sitting).
If you are often using the mattress for sitting as well then edge support would probably be more important to you so I would make sure that you test for this as well.
As with any mattress purchase I would make sure that your mattress uses good quality and durable materials … especially in the upper comfort layers … so that the areas you sit in most frequently don’t soften or break down too quickly.
Foam mattresses will tend to be more “point elastic” which means that they can sink in more deeply with the more concentrated weight of sitting so I would make sure that you include some innerspring mattresses in your research so that you can see and feel the difference between firmer innersprings and mattresses that use either firmer polyfoam or latex in the support core of the mattress.
You will likely need a support core (innerspring or polyfoam or even latex depending on your preferences) that is a little firmer than normal to keep you from sinking in too much when you are sitting and then you will need comfort layers that are “just enough” in terms of thickness and softness so that they will relieve pressure points in all your sleeping positions when you are lying down and sleeping on the mattress so that you don’t sink into the mattress too much when you are sitting.
I would also keep the height of your daybed and mattress in mind so that the surface of your mattress isn’t too high or too low for you to sit comfortably.
If you let me know your city or zip code I’d also be happy to let you know about any of the better options or possibilities I’m aware of in your area.