From the Wikia article on Gotei 13:
Lieutenants function as executive officers for their division, taking care of or supervising day-to-day operations. Along with their captains, they are not assigned to one specific squad. They generally only know the Shikai of their Zanpakutō but they are still the second strongest in their division. In the case of a division captain's death, departure, or other circumstances making him unable to perform their duties, the lieutenant acts as the "substitute captain" until another can be assigned.
The authority to appoint or dismiss a lieutenant lies solely with the captain of the respective division, a privilege which was granted to them with the support of the Central 46 Chambers. Captains reserve the right to appoint multiple lieutenants at a single time, though this is rarely done due to it being both considered highly unorthodox and heavily frowned upon by the Central 46 Chambers.
Hence, it's not necessary for a Lieutenant to have a Bankai. It's not even required for the second strongest person of the squad to be the lietenant as it is completely upto the captain to the assign the position, although the position is usually given to the second strongest person in the squad.
An important factor in Aizen deciding to appoint her as a lieutenant has to do with the fact that she was in love with him, which made it easy for him to manipulate her.
From the Wikia article on Momo Hinamori:
Though normally kind, her rare bouts of anger can appear rather menacing, such as her outrage at Captain Gin Ichimaru when she blamed him for killing Aizen. While serving as his lieutenant, Momo greatly respects and admires Aizen. She idolizes him to the point where a letter from Aizen leads her to attack Hitsugaya, her childhood friend, and even after being stabbed by Aizen, she, proclaiming Aizen was being manipulated, asks Hitsugaya to save him. When Aizen leads an attack on the Fake Karakura Town, Momo helps to defend the town from Aizen and his forces, but she still notably refers to him as "Captain Aizen".
Aizen himself says:
When she is reunited with Aizen, he, betraying her, stabs her through the chest, and she nearly dies.
Aizen's reasoning behind almost killing Momo is since he built her to not be able to live without him, he has done her a favor by killing her
Even though she might be weak, she does have battling skills. You can read about her powers and abilities here.
In the end, we can say that her appointment to the position of a vice-captain/lieutenant has more to do with her being of use to Aizen than on the basis of her fighting prowess.