Are there any mantras/stotras that a parent can chant for the well being (health/wisdom/ long life etc) of their children?
Yes, there are.
The presiding deity of children is said to be Goddess Shasti. She is Devasena--the consort of Lord Karthikeya.
In Bengal, many mothers fast during the Shasti Vrata for ensuring protection and general well-being of their children.
The Devi Bhagavatam contains separate section in it that describes her Mantra, Stotra and also gives the story of how her worship commenced on the earth.
- Nârada said :--“O Lord! Sasthî, Mangalâ Chandî, and Manasâ, are the parts of Prakriti. Now I want to hear the lives of them.” 4-22. Nârâyana said :-- O Child! The sixth part of Prakriti is named as Sasthî. The Devî Sasthî is the Presiding Deity of infants and children; She is the Mâyâ of Visnu and She bestows sons to all. She is one of the sixteen Mâtrikâs. She is known by the name of Devasenâ. She performs Vratas (vows); She is the chaste and dearest wife of Skanda. She decides on the longevity of children and is always engaged in their preservation. So much so, that this Siddha Yoginî always keeps the children on her side.
A story involving king Dharmadeva and Priyavrata is given which provides the details of how the worship of Shasti Devi began on earth. The Mantra of Shasti Devi is also given as well as a Stotra.
Here are some more relevant verses:
On the twenty-first day, the auspicious moment, at the ceremony of giving rice to a child for the first time, when sixth months old, and on all other auspicious ceremonies of the children, Sasthî Devî’s worship was made extant and the king himself performed those worships with great care and according to due rules. Now I will tell you about the Dhyânam and method of worship and stotra as I heard from Dharma Deva, and as stated in Kauthuma S’âkhâ. Hear. He has said :-- In a S’âlagrâma stone, in a jar, on the root of a Bata tree, or drawing the figure on the floor of the rooms, or making an image of Sasthî Devî, the sixth part of Prakriti and installing it, one should worship the Devî. The Dhyânam is this :-- “O Devasenâ Thou art the bestower of good sons, the giver of good luck; Thou art mercy and kindness and the progenitor of the world; Thy colour is bright like that of the white Champaka flowers. Thou art decked with jewel ornaments. Thou art pure, and the highest and best Devî. Obeisance to Thee! I meditate on Thee.” Thus meditating, the worshipper should place flower on his own head. Then again meditating and uttering the principal mantra one is to offer the Pâdya (water for washing feet), Arghya, Âchamanîya, scents, flowers, dhûp, lights, offerings of food and best roots and fruits and one should worship thus with various things Sasthî Devî. “Om Hrîm Sasthî Devyai Svaha” is the principal Mantra of Sasthî Devî. This great Mantra of eight letters a man should repeat as his strength allows. After the Japam, the worshipper should chant hymns with devotion and undivided attention and then bow down. The Stotra (hymn) of Sasthî Devî as per Sâma Veda is very beautiful and son-bestowing.The lotus-born Brahmâ has said :-- If one repeats (makes Japam) this eight lettered mantra one lakh of times, one gets certainly a good son. O Best of Munis! Now I am going to say the auspicious stotra of Sasthî Devî as composed by Priyavrata. Hear.
The Goddess can be prayed to if one wants good children (if they don't have any) and also for seeking protection and well-being of the already existing children.
The Phalasruti of the aforementioned Stotra ends thus:
Through the grace of Sasthî Devî, that son becomes a hero, well qualified, literate, renowned and long-lived. If any woman who bears only a single child or delivers dead children hears with devotion for one year this stotra, she gets easily, through the Devî’s grace, a good son. If the father and mother both hear with devotion, this story during the period of their child’s illness, then the child becomes cured by the Grace of the Devî.
Source for all the verses given above is Devi Bhagavatam, Book 9, Chapter 46.
You can get the full Stotra (for the purpose of recital) in the above mentioned Chapter of the Purana.
I have found a Shasti Devi Stotram, that is found in Subhramanya Vaivrutha Purana (I have never heard about this Purana before though), on Hindupedia.com.
This website, however, claims that the Stotra is from the Brahmavaivarta Purana.
Here are some vedic Mantras from Atharva Veda that pray's various deities for well being of their children. Parents can also use these mantras through them or after proper consultation with priests or acharyas as these are vedic Mantras.
Atharva Veda Kanda 2 - Sukta 13 - Mantra 4 - Dirghayu Sukta.
This Mantra is chanted at the time of ceremony of wearing first garments to new born baby by parents. Here the Mantra pray to various powers of gods so that child could live long life (upto 100 years) .
एह्यश्मानमा तिष्ठाश्मा भवतु ते तनु : |
कृण्वन्तु विश्वे देवा आयुष्टे शरद : शतम ||4||
O child come sit firmly on this rock ,so that your skin will become firm as rock. May the godly powers make your age of 100 years.
Atharva Veda Kanda 2 - Sukta 28 - Mantra 5 - Dirghayu Prapti Suktam.
Here In this Mantra Agni Deva , Mitra-Varuna ,Aditi and Vishwa- Devas are asked to bless a child with various good qualities.
इममग्न आयुषे वर्चसे नय प्रियं रेतो वरुण मित्रराजन |
मातेवास्मा आदिते शर्म यच्छ विश्वे देवा जरदष्टिर्यथासत् || 5||
O God Agni give this child a life of 100 years and make him lustrous
(Tejasvi). O Mitra -Varuna make this child fit to give birth to his children.O Aditi Devi bless this child with utmost joy like a mother. O Vishwa-Devas make (bless) this child with special qualities and abilities.
Atharva Veda Kanda 2 - Sukta 36 - Mantra 1 -Pativedan Sukta.
Here this sukta prays to lord agni so that the girl child could get a husband who is wealthy and who is full of good qualities and prosprous. Here we can see parents concern / Prayer for marrige of unmarried girl.
आ नो अग्ने सुमतिं संभलो गमेदिमां कुमारीं सह नो भगेन |
जुष्टा वरेषु समनेषु वल्गुरोषं पत्या सौभगमस्त्वस्यै || 1||
O Agni let our this intelligent unmarried girl get a husband who is full of good qualities ,who is wealthy and prosperous. Let this girl child gets a fortune to live life with her husband. i.e. let she get married.