Yes, I think it was worthy of a Best Picture nomination. It's a reasonably sophisticated and entertaining "family comedy," and it was a very significant film that had a major impact on Hollywood at the time.
It was the first film to take a chance on what we now refer to as a "tween" star, at a time when Hollywood couldn't see any potential in an adolescent girl as a potential star. With all the Hayley Millses' Hillary Duffs, Miley Cyruses and other young adolescent stars who have come up in the years since, this might seem inconcievable, but in the 1930s, studios avoided adolescent actresses as stars in their own right like the proverbial plague. More than any other adolescent actress at the time, Deanna Durbin and the enduring success of her films for Universal, changed all of that.
It was an enormous hit, both critically and financially, landing on the practially every "Ten Best List" of films for that year, and was one of the few films to be proclaimed a hit (and its' leading lady an instant star) based on preview screenings. It also helped to save Universal Studios from bankruptcy, and it's success (and the success of other Durbin films) inspired all the other studios (and Universal, too) to develop other "teen" actresses like Judy Garland, Gloria Jean and Bonita Granville. And, since Durbin was a classically-trained vocalist, it helped to introduce younger viewers to classical music and gave adolescent viewers a star with whom they could identify.
Its' innovations have been so imitated over the years that it seems somewhat dated now, but at the time it was considered fresh and original, even cutting edge, in many ways.