What would make a mom proud of her child in medieval times?

My answers relate primarily to northern Western European Christians. (I'm not familiar enough with Italian women's writing to cite examples from that corpus; my apologies.)

There are two types of source material to turn to here: looking at the general responsibilities of mothers in childraising, and the words of women themselves. In the second category, we're unfortunately very limited because most surviving women's writing from the Middle Ages (especially earlier on) comes from celibate nuns, but there are a couple of really interesting examples.

Clerics commenting (or prescribing) child-raising practices generally agree that (peasant/gentry once there was a gentry) women were responsible for all children until age 7, and then the girls continued to spend the days with their mothers learning how to be a farmwife or a woman in the family trade. As literacy and clerics' attention to the religious knowledge of the laity increased in the late Middle Ages, sermons exhorted women to teach their children to read, as well as instruct them in the basics of Christian doctrine and morality. We can infer from that that a good mother was supposed to take pride in raising moral, Christian children who were good at their labors. It's hard for me to imagine a mother not also being proud that her son or daughter won the Carnival footrace or sewed the best ABC sampler.

The question of joining religious life is complicated. Certainly the literature is full of families who placed a heavy importance on having a son or daughter (or more than one) take vows, with certain orders and houses seen as more prestigious depending on the place and era. This would be for both status and pious reasons. However, that was very often a question of parental initiative rather than offspring's choice (even in cases not of child oblation). In this case the placement, rather than an achievement, would be the source of pride. The same for attaining a status like abbess in a house or advancing through clerical orders: that could be the result of family machinations and donations rather than a child's efforts and popularity within their community, but still a very desirable goal for the family.

There is a fairly strong hagiographical topos of saints' parents disapproving of their super-religious lifestyles, although that is certainly a very different degree of religious devotion than singing psalms in choir!

As far as women's own writing:

Dhuoda was a 9th-century Frankish noblewoman who actually wrote a life handbook for her son, called the Liber manualis. Much of the work falls under instruction in Christian doctrine. As far as behavior, she emphasizes charting the right course of virtue, the importance of chastity (which in the Middle Ages, can mean chastity before marriage and fidelity to the spouse once married), and obedience to his patriarchs: king, lord, father. "You can discover what [rules] you ought to follow for my sake"? That sounds like a description of what would make Dhuoda proud as a mother.

Second example, Christine de Pisan, Italian-by-birth noblewoman in 14th-15th century France. Her laments about the sorry state of education for girls (and composition of books partially to remedy that) suggests that she placed a high premium on her daughter's own learning. She was also very pleased that her daughter attained a place in an important convent.

Example from the very end of the Middle Ages/early Reformation, a 15th-16th century German noblewoman named Argula von Grumbach. In letters to and about her children, she was very proud of one of her sons who apparently did very well at his studies (religious and otherwise), and expressed her vexation and disappointment with the son who ~~struggled~~ was less concerned with learning than getting into serious trouble with his friends.

I'm not very familiar with medieval Jewish culture, but I'll give it a shot. Jewish women were also primary caretakers of children under seven. Jewish fathers were responsible for religious education; however, Elisha Baumgarten notes that the amount of halakhic commentaries ranting on this topic suggests that in practice, women took some role in religious education and men took some role in other areas. Additionally, a theme relayed over and over in medieval Jewish martyr literature is mothers killing their own children rather than having them seized and baptized by Christians. I think it's safe to infer Jewish women were proud of children who remained Jewish under what could be, um, a lot of pressure to convert.

And this is to say nothing of spiritual motherhood...

Major sources: Shulamith Shahar, Childhood in the Middle Ages. Elisha Baumgarten, Mothers and Children: Jewish Family Life in Medieval Europe. Marcelle Thiébaux, ed., Dhuoda: Handbook for Her Warrior Son (English and Latin facing-page). Peter Matheson, ed., "Argula von Grumbach: A Life in Letters" (Early Modern Women 4; German with English translations). Christine de Pisan...take your pick?

ETA Dhuoda's century.