Here we're dealing with two basic handicaps: the still-primitive state of research on medieval Muslim women, and the mind-boggling expanse of the medieval Islamic world in space and time. But while it's probably even harder to talk about the medieval Muslim women than the medieval Christian woman, focused studies can help sketch out some possibilities and even potential patterns.
Most basically, Islamic law guarantees women the right to inherit, own, and discharge property, including independently in marriage. Property could include land, cash, stuff, and slaves, which already points out some of the basic ways Muslim women involved themselves in the monetized economy. (To distinguish from unpaid labor in the household).
Women who inherited land, not uncommon in Anatolia and the Levant, acted as keen managers and landlords. Irini bint Yorgi inherited an additional house in Herekuna along with an orchard and a separate garden. Income from rent and from the sale of produce helped swell her own coffers. In Turkish territory, women appear in court cases related to buying and selling livestock, too.
Women also seized the chance to make their cash property work for them. In al-Andalus continuing into Christian rule over Iberia, Muslim women sometimes acted as moneylenders. Gumus Hatun took it a step further: she was part of a twelve-person trading company. In this realm, however, women's expenditure is much more celebrated in the sources than income: women played critical roles in religious patronage of mosques, madrasas, wells and shelters along pilgrimage routes, and Sufi lodges. But in some cases, this helped provide an economic opening for other women: women did indeed act as religious teachers in the Near East.
A common theme in normative or prescriptive sources is that Muslim women must stay secluded in their homes. In practice, we know this was (a) limited to upper class women, and (b) upper class women still often found ways around it. We can follow scholars' grumpiness to find women in markets from Spain to Mali to Baghdad, with authority Yahya complaining that sure, women veiled their faces and bodies adequately to maintain seclusion, but they wore flip-flops in order for the noise to make men notice them anyway. I generally see textiles (woven fabric, constructed items, and embroidered/decorated items) and food as noted things that women sold in their market stalls, especially in Egypt.
Entering into the realm of labor, from Christian Iberia, we know Muslim (and Christian and Jewish) women used communal spaces for laundry and baking bread. This probably acted as a way for some enterprising women to enter into the business world: in court cases where a slave is accused of ruining laundry, which occur in Turkey as well, the defense is often that "I paid someone else to do it."
One of the most important and female careers, of course, was midwife. Some Muslim women also worked for money as wet nurses, although in Spain, this seems to have been a more common role for slaves. And even though slavery was a key domestic institution in the medieval Mediterranean world (Christian as well as Islamic), Muslim women also worked as paid servants in other homes.
And then, of course, there were the less reputable ways to support oneself. Some Muslim women resorted to prostitution--in al-Andalus, there were actually a few mosques that developed a reputation as basically a courtyard where men could bring a prostitute without people looking over his shoulder. For women, prostitution sometimes went hand-in-hand with a career in performing music. The "singing girls" of caliphs' and high nobles' courts are both the most famous and usually slaves (still, skill with poetry and instrument could absolutely elevate one above another), but singers or "singers" were hired by the more ordinary folk for celebrations and parties.
And of course, women stole, including from each other. There's a famous case from 16th century Uskudar: a woman named Nar Hatun just wanted to enjoy the steambaths after a rough day traveling. But when she returned to get dressed and claim her things, she was missing her hat, her shirt, and her towel. Another bathee, it turned out, had seized upon the fineness of the fabrics to make some material gain through theft.
Economic activities of medieval Muslim women, you might say, both uncover and are uncovered by the gap between one religious ideal world where women were associated with family, household, and domestic life to the extent that they were not supposed to be seen outside it; and the other religious ideal world where women were key parts of an economically successful society.