C++ function in parent return child

You should use the CRTP.

template<class Derived>
class Entity
{
    Derived* This() { return static_cast<Derived*>(this); }

public:
    Derived& name(const string& name)
    {
        ...
        return *This();
    }
};

class Client : public Entity<Client>
{
public:
    Client& budgetRange(const long& range)
    {
        ...    
        return *this;   
    }
};

If you want to use virtual functions, you can also add abstract base class, like this:

class AbstractEntity
{
public:
     virtual void foo() = 0;

     virtual ~AbstractEntity();
};

template<class Derived>
class Entity : AbstractEntity
{...};

The "curiously recursive template" pattern could help here; make the base class a template, parametrised by the derived class, along the lines of:

template <typename Derived>
struct Entity {
    Derived & name(std::string const & name) {
        // stuff
        return static_cast<Derived&>(*this);
    }
};

struct Client : Entity<Client> {
    Client & budget(long range) {
        // stuff
        return *this;
    }
};

Client().name("Mike").budget(50); // should compile

This will only work if all your types inherit directly from Entity. If you need the types to be polymorphic (i.e. all share a common base class), then you'll need to add another non-template base class, and have Entity inherit from that.

Now that nearly everything has already been said, I want to add a piece of answer that allows one to use the CRTP over multiple levels of inheritance:

The above CRTP implementations break when one wants to inherit from Client, since Derived will refer to Client. In case you want to be able to carry the named parameter idiom over multiple levels of inheritance using CRTP pattern, you need to code your classes like so

template<class Derived>
class Entity_T
{
protected:
    Derived* This() { return static_cast<Derived*>(this); }
public:
    Derived& name(const string& name)
    {
        ...
        return *This();
    }
};

template<class Derived>
class Client_T : public Entity_T<Derived>
{
    Derived& budgetRange(const long& range)
    {
        ...    
        return *This();   
    }
};

To provide the user a template-free version of Client_T add

class Client : public Client_T<Client> {};

Whether or not this is worth the enlarged code base is entirely up to you. Note, that I haven't compiled the code above.