'object' does not contain a definition for dynamic

It is not advisable to use dynamic object in your use case. But this is my opinion.

Anyway, to access member of dynamic object,

string fileName = couponData.GetType().GetProperty("LogoName").GetValue(couponData, null);

"RuntimeBinderException" has already been answered on below articles please refer it.

https://social.msdn.microsoft.com/Forums/en-US/30b916bf-7e59-4d8d-b7bc-076d4289a018/type-inference-turns-my-vars-to-dynamic?forum=csharplanguage

Try to below code:

public static dynamic GetCouponDetailsbyCouponID(Guid couponID)
{
    using (var loEntities = new Entities())
    {
        var nonWinnerGift = (from nw in loEntities.CorporateNonWinnerGift
            join um in loEntities.Users on nw.UserID equals um.Id
            where nw.IsDeleted != true && nw.CouponID == couponID
            select new
            {
                FullName = (um.FirstName + " " + um.LastName),
                Title = nw.Title,
                Description = nw.Description,
                LogoName = nw.LogoName,
                CouponID = nw.CouponID,
                IsDiscount = nw.IsDiscount,
                Discount = nw.Discount,
                Desclaiemer = nw.Desclaiemer

             }).SingleOrDefault();

        dynamic d = new ExpandoObject();

        d.FullName = nonWinnerGift.FullName;
        d.Title = nonWinnerGift.Title;
        d.Description = nonWinnerGift.Description;
        d.LogoName = nonWinnerGift.LogoName;
        d.CouponID = nonWinnerGift.CouponID;
        d.IsDiscount = nonWinnerGift.IsDiscount;
        d.Discount = nonWinnerGift.Discount;
        d.Desclaiemer = nonWinnerGift.Desclaiemer;

        return d;
    }
}