Yes, indeed, the repetition of her name does have significance; Jesus utters no word in vain.
St. Thomas Aquinas, in his Catena Aurea (Golden Chain) of the Fathers' commentaries on the Gospels, reports what St. Augustine wrote commentating on Luke 10:38-42:
AUG. Martha was as well engaged in ministering to the bodily wants or wishes of our Lord, as of one who was mortal, but He who was clothed in mortal flesh, in the beginning was the Word. Behold then what Mary heard, The Word was made flesh. Behold then Him to whom Martha ministered. The one was laboring, the other at rest. But yet Martha, when much troubled in her occupation and business of serving, interrupted our Lord, and complained of her sister. For it follows, And said, Lord, do you not care that my sister has left me to serve alone? For Mary was absorbed in the sweetness of our Lord’s words; Martha was as preparing a feast for our Lord, in whose feast Mary was now rejoicing. While then she was listening with delight to those sweet words, and was feeding on them with the deepest affection, our Lord was interrupted by her sister. What must we suppose was her alarm, lest the Lord should say to her, “Rise, and help your sister?” Our Lord therefore, who was not at a loss, for He had shown He was the Lord, answered as follows, And Jesus answered and said to her, Martha, Martha. The repetition of the name is a mark of love, or perhaps of drawing the attention, that she should listen more earnestly. When twice called, she hears, You are troubled about many things. that is, you art busied about many things. For man wishes to meet with something when he is serving, and can not; and thus between seeking what is wanting and preparing what is at hand, the mind is distracted. For if Martha had been sufficient of herself, she would not have required the aid of her sister. There are many, there are diverse things, which are carnal, temporal, but one is preferred to many. For one is not from many, but many from one. Hence it follows, But one thing is needful. Mary wished to be occupied about one, according to that, It is good for me to cling close to the Lord. The Father, the Son, the Holy Spirit, are one. To this one he does not bring us, unless we being many have one heart.