We've been through this - its terrible to see your precious baby with eczema and itching all the time. Unfortunately every baby is different (it's so annoying when people say that!!) but this is what worked for us.
I tried all of the different things on prescription: diprobase (useless), liquid paraffin (made my daughters skin feel like a roasted chicken), aqueous cream (didn't seem to do anything and I was terrified of what the Internet says about it) etc etc.
We were using hydrocortisone every day all over her as well.
In the end I did what a friend recommended and ordered some Aveeno Eczema Baby cream off of amazon. They don't sell it in the uk so it had to be shipped from the us and takes AGES. But, it's the only thing that cleared up her skin. For us, it's been amazing. Now I buy the Aveeno Skin Therapy and though it doesn't work quite as well it does the job and its loads cheaper and available here. I still use the steroid occasionally to spot treat little bits that are getting out of hand. We use dermol 500 as soap in the bath and balneum in the water - both on prescription.
What laundry detergent are you using? My LOs skin reacts to detergents and we have found that the only one we can use is a brand called SImply. We have used both Simply Pure and Simply Sensitive and they both work fine. Again, it's hard to find (of course!). We have also found that we have to wash all of our clothes and bedding etc in it as well - great because it doesn't actually clean very well!!
My LO is allergic to dairy, egg and sesame (confirmed with a skin prick test last week but we always suspected it). Cutting out dairy and egg from my diet (we are breastfeeding) helped a lot. If you are formula feeding, maybe consider asking for a skin prick test if you can't sort out the eczema. (And don't let them fob you off with the "lots of babies have eczema/they will grow out of it/you are being over anxious" bs!). They can prescribe hypoallergenic formula if needed.
It's worth educating yourself about the atopic march. Basically there is a strong connection between eczema, allergies and athsma. Sigh...the things we never thought about before we became moms...
Hope your little one's skin clears up soon.