Just for fun, here's a fairly big number of fixes. I'm not going to enumerate them all. basically:
- pass fixed size arrays by reference, to avoid unsafe usage
- check input errors rigorously, and do it in a single loop
- several misguided array assignments that either did nothing or prevented the result being seen by the caller...
- use charfor the input type (soQwould ever get matched) and prefer aa switch over repeated ifs
- See below for a more typical C++ style solution, including a check for WIN or LOSE
Exercise for the reader: seed your rand() generator, or it was pretty simple to win this lottery (pasting in the previous winning ticket :))
Output on a demo run (shows error handling as well):
LITTLETON CITY LOTTO MODEL:
---------------------------
 1) Play Lotto 
 Q) Quit Program 
Please make a selection: 1
 Please enter your name: me
 Please enter number 1:34
 Please enter number 2:7
 Please enter number 3:16
 Please enter number 4:18
 Please enter number 5:24
 Please enter number 6:24
Duplicate entry 24
 Please enter number 6:7
Duplicate entry 7
 Please enter number 6:37
 Please enter number 7:whoops
Bad input 'whoops' discarded
 Please enter number 7:what was the last number again?
Bad input 'what' discarded
 Please enter number 7:Bad input 'was' discarded
 Please enter number 7:Bad input 'the' discarded
 Please enter number 7:Bad input 'last' discarded
 Please enter number 7:Bad input 'number' discarded
 Please enter number 7:Bad input 'again?' discarded
 Please enter number 7:27
User ticket:    7; 16; 18; 24; 27; 34; 37; 
Winning ticket: 7; 16; 18; 24; 27; 34; 36; 
Result of draw: WINNER
.
#include <string>
#include <iostream>
const size_t amount = 7;
void getLottoPicks(int (&ticket)[amount])
{
    for(int i = 0; i < amount; i++)
    {
        do
        {
            std::cout << "\n Please enter number " << i + 1 << ":";
            if(std::cin >> ticket[i])
            {
                if(ticket[i] >= 1 && ticket[i] <= 40)
                {
                    bool ok = true;
                    for(int check = 0; ok && check < i; check++)
                    {
                        ok &= (ticket[i] != ticket[check]);
                    }
                    if(ok)
                    {
                        break;
                    }
                    else
                    {
                        std::cout << "You cannot use the same lotto number twice. Try again." << std::endl;
                    }
                }
                else
                {
                    std::cout << "Your selection cannot be less than 1 and/or greater than 40. Try Again." << std::endl;
                }
            }
            else
            {
                std::cin.clear();
                std::string discard;
                std::cin >> discard;
                std::cout << "Bad input '" << discard << "' discarded";
            }
        }
        while(true);
    }
}
void GenWinNums(int (&winningNumbers)[amount])
{
    for(int i = 0; i < amount; i++)
    {
        winningNumbers[i] = (rand() % 40) + 1;
        while(winningNumbers[i] < 1 || winningNumbers[i] > 40)
        {
            winningNumbers[i] = (rand() % 10) + 1;
        }
        if(i > 0)
        {
            for(int check = 0; check < i; check++)
            {
                while(winningNumbers[i] == winningNumbers[check])
                {
                    winningNumbers[i] = (rand() % 10) + 1;
                }
            }
        }
    }
}
int main()
{
    std::string name;
    char choice;
    std::cout << "LITTLETON CITY LOTTO MODEL:" << std::endl;
    std::cout << "---------------------------" << std::endl;
    std::cout << " 1) Play Lotto " << std::endl;
    std::cout << " Q) Quit Program " << std::endl;
    std::cout << " Please make a selection: ";
    std::cin >> choice;
    switch(choice)
    {
        case '1':
            {
                std::cout << "\n Please enter your name: ";
                std::cin >> name;
                int user_picked[amount];
                int chosen_lotto[amount];
                getLottoPicks(user_picked);
                GenWinNums(chosen_lotto);
                std::cout << "\n";
                for(int i = 0; i < amount; i++)
                {
                    std::cout << user_picked[i] << "; ";
                }
                std::cout << "\n";
                for(int i = 0; i < amount; i++)
                {
                    std::cout << chosen_lotto[i] << "; ";
                }
            }
        case 'Q':
        case 'q':
            return 0;
        default:
            std::cout << "\n That is not a valid choice." << std::endl;
    }
    std::cout << "\n Press enter...";
    std::cin >> choice;
}
More typical C++ solution
#include <string>
#include <iostream>
#include <algorithm>
#include <iterator>
#include <set>
const size_t amount = 7;
typedef std::set<int> ticket_t;
ticket_t getLottoPicks()
{
    ticket_t ticket;
    while (ticket.size() < amount)
    {
        std::cout << "\n Please enter number " << (ticket.size()+1) << ":";
        int entry;
        if(std::cin >> entry)
        {
            if(entry >= 1 && entry <= 40)
            {
                if (ticket.end() != ticket.find(entry))
                    std::cout << "Duplicate entry "  << entry;
                else
                    ticket.insert(entry);
            } else
                std::cout << "Entry out of range [1..40]: "  << entry;
            continue;
        }
        std::cin.clear();
        std::string discard;
        std::cin >> discard;
        std::cout << "Bad input '" << discard << "' discarded";
    }
    return ticket;
}
ticket_t genWinNums()
{
    ticket_t ticket;
    while (ticket.size() < amount)
        ticket.insert(rand() % 40 + 1);
    return ticket;
}
std::ostream& operator<<(std::ostream& os, ticket_t const& t)
{
    std::copy(t.begin(), t.end(), std::ostream_iterator<int>(os, "; "));
    return os;
}
int main()
{
    std::cout << "LITTLETON CITY LOTTO MODEL:" << std::endl;
    std::cout << "---------------------------" << std::endl;
    std::cout << " 1) Play Lotto "             << std::endl;
    std::cout << " Q) Quit Program "           << std::endl;
    std::cout << "Please make a selection: ";
    char choice;
    std::cin >> choice;
    switch(choice)
    {
        case '1':
            {
                std::cout << "\n Please enter your name: ";
                std::string name;
                std::cin  >> name;
                const ticket_t user    = getLottoPicks();
                const ticket_t winning = genWinNums();
                std::cout << "User ticket:    " << user    << "\n";
                std::cout << "Winning ticket: " << winning << "\n";
                // check win?
                bool ok = 0 == std::lexicographical_compare(
                        user.begin(), user.end(),
                        winning.begin(), winning.end());
                std::cout << "Result of draw: " << (ok? "WINNER":"No luck") << "\n";
            }
        case 'Q': case 'q':
            return 0;
        default:
            std::cout << "\n That is not a valid choice." << std::endl;
    }
    std::cin.ignore(1<<24,'\n');
    std::cout << "\n Press enter...";
    std::string dummy;
    std::getline(std::cin, dummy);
}