I would say the main qualities that one should emphasize is thoroughness, an attention to detail, and versatility. Gaining paralegal positions is about reassuring the attorneys that your work product will need relatively little review/proofing time, and that you can be relied upon to quickly change projects should an urgent deadline arise.
In order to be successful as a paralegal you must have excellent organizational skills, pay attention to details, and manage time well.
Your organizational skills will keep files in order, documents provided to the proper client, and preserving evidence or research. A paralegal often keeps track of all of the attorney's appointments and provides her with everything she needs each month, week, day and hour. (It's like mind reading, you are guessing her needs before she needs it.)
Attention to detail, the big details and the little details. When drafting form documents you want to be certain all blanks are filled. You want to be clear and precise in directions to clients.
Organizing your time comes in handy all the time. You want to make sure your work is done in a timely fashion. Legal documents and legal timelines and statutes of limitations do not care that you are having an off day, you do not want to miss deadlines. Often during a day I had my day mapped out and then we stopped everything to collaborate on a large legal brief that had an immediate deadline. So do your most time sensitive projects first because you will get interrupted.
Depending on the kind of law your office practices you might deal more with clients. Most of these people are having a rough time and may be scared about the process they are going through. Genuine empathy is a great quality in a law firm.
Working as a Legal Assistant/Paralegal has been one of the best experiences in my life. working closely with attorney made me realized how important is to have a good customer service. People who are looking for an attorney feel vulnerable and at risk, that is why attorney love Legal Assistants that are good at talking to people and obtaining the information needed without making feel a client pressured or judge.
I would say that it is important to highlight good communication and problem solving skills. In addition, attorneys are always busy and with a thousand things on their mind, therefore it is important to let them know during the interview that you're willing to learn things by yourself and to jump into any chaos and do everything possible to solve it. It is really important to have initiative in this field since as a legal assistant you might now what will be next in the process and starting to work on things early will facilitate the attorney's life so they will love that. Being detail oriented is also important because before a judge, a mistake can be the end of a case.