[part 1]:

Hello, sorry for the really late response.

Tl; dr: If we really limits the period in question to the 11th century (Late Viking Age), surprisingly enough, women maybe have enjoyed nearly equal access to literacy as men in contemporary Scandinavia - mainly due to the limited local option to access there.

First of all, researchers suppose now suppose at least two modes of literacy, namely "runacy (runic literacy)" and that of Latin alphabet co-existed side by side from the late 10th century to the High Middle Ages in Scandinavia as well as North Atlantic Isles (Spurkland in Hermann 2005).

Thus, I'll deal with these two modes of literacy separately in individual sections below.

1: Women and Runic Alphabets (fuþark) (and their Use) in the 11th Century

Runic inscriptions are generally counted one of two "indigenous contemporary primary sources" from the Viking Age Scandinavia (another is a skaldic poetry that the poets composed in Old Norse with very complicated metrics and kennings (metaphoric allusions by the combinations of the words))
, and the 11th century was indeed the heyday of the practice of erecting rune stones (Sawyer 2000). Scandinavians in the late Viking Ages are also known to have carved runic scripts on different materials (more than different kind of stones) like wooden sticks, however.

Concerning more details on the wide aspects of this kind of "runacy", I'd also recommend you to check /u/Platypuskeeper's previous posts in What were Norse/Germanic runes used for?.

Among about 3,000 runic stones, Sawyer estimates about 12% of the total extant stones are commissioned by the woman (only), and further 15% are by the joint commission between the male and the female patrons (Sawyer 2000: 111-13). As Sawyer herself emphasizes, however, these figures, especially the latter, can be not so absolute due to the regional variations of the extant materials (their concentration on central Sweden). She also associates this regional concentration of the joint-commissioned rune stone with the relative weakness of the emerging royal authority as well as possibly different property inheritance pattern that enabled the woman to inherit the property and to act as a patron for the wealth-displaying commission of the ornamented rune stone. Anyway, it is rather natural to suppose from these figures that not a small number of women, at least of good social status [enough to commission the runic stone], could probably read runes and also decide the content of the inscription by herself.

I also cited the example of the female patron, royal mother Asfrid (Ásfríðr in Old Norse) who lived in now southern Denmark in the early 10th century before in: Who were the ruling class of the Viking Age in Scandinavia? Were they chosen by merit, or born into powerful families?:

• DR2, called large Sygtrygg Stone states in the inscription: "§A Ásfríðr made these monuments in memory of Sigtryggr, §B her son and Gnúpa's (English translation is taken from Runor App.)."
• DR4, called small Sygtrygg Stone also states in the inscription: "§A Ásfríðr, Óðinkárr's daughter, made these monuments in memory of King Sigtryggr, §B her son and Gnúpa's. §C Gormr carved the runes."

We "unfortunately" know little how the Viking Age Scandinavians learned runic scripts, but "fortunately" regardless of gender - the "carver" mentioned above [Gormr in DR4] was the quasi-professional (sometimes perhaps a relative by blood, though) to carve them on behalf of the patron/ commissioner (Källström 2007). Therefore, the fact that both male and female local elites in late Viking Age Scandinavia commissioned the runic stone would probably be suffice to suppose the spread of "runacy" also among women.

Some might argue against what I've said above, by the absence of the actual female carver/ reader of runic scripts, however. To counter their argument, though not exactly contemporary, I introduces two inscriptions (or graffitis) from the Old Norse spoken areas in the High Middle Ages.

A band of Norwegian-Orcadian crusaders once broke into the neolithic burial mound, Maeshowe (called Orkahagr (the mound of Orkney) in later saga of the earls of Orkney (Orkneyinga saga)) in the middle of the 12th century and carved some runic graffitis on the chamber's wall. Among them, we come across the following text:

"§A Jerusalem-travellers broke Orkhaugr. Hlíf, the earl's houskeeper, §B carved (Or Barnes 24)."

This carver Hlíf certainly has a female name, and apparently of low social standing (servant of the earl's household of Orkney).

Another example is actually a marriage proposal letter, carved on a part of wooden walking stick around 1300 and found on the floor of the church in central Norway around 1300 CE!:

"§A Hávarðr sends Guny/Gunnhildr(?) his friendship and Gods greeting. §B And now it is my full desire to ask you (for your hand in marriage), if you do not want to be with §C Kolbeinn. Think over your intentions … and have me told §D your desire (N A 74)."

Spurkland imagines several possible scenarios on the background of this inscription (Spurkland 2005: 196-99), and his likeliest scenario is that Gu(ð)ny rejected the proposal in favor of another suitor, Kolbeinn, and threw Hávarðr's wooden block with this runic inscription away while her visiting in the church as soon as possible......Regardless of which story, however, the author of the inscription clearly presuppose that the intended female receiver of the stick, Guny/Gunnhildr, could read the runic text by herself in a relatively short time.

Since we don't have any positive evidence that the acceptance of Christianity encouraged the spread of this kind of "runacy" further among women, it would inevitably be natural to suppose that wider social strata of women could also know how to read and "write" (rather carve) runic inscription if necessarily and with a proper tool like a knife (for the wooden tablet/ stick) already in the 11th century.

2: Literacy [of Latin Alphabets], Availability of Education especially for Women in the 11th Century Scandinavia

Well, really thank you for OP's patience (especially if OP is primarily interested only in this kind of literacy).

Some basic premises are:

• Christian clergies introduced Latin scripts (with Christianity) in Scandinavia.
• Around 11th century, Scandinavian society accepted Christianity and saw the establishment of Christian Nordic Kingdoms (though I place that of Sweden rather later), usually allied with the clergies (though not always with all the visiting clergies).
• Cathedral schools, monasteries and sometimes court chapels functioned as educational institutions in Early (Contemporary) Medieval West.
• The oldest extant document written by Latin scripts and we know the exact text dates in Scandinavia only back to 1085 CE (royal donation charter by King Knud den Hellige of the Danes to the cathedral of Lund and copied later into the cathedral chapter's necrology), and if we strictly limits the original (the document itself), the oldest document dates further only back to 1135 CE (Ozawa 2009).

Taken these premise into consideration together, the access to the educational institutions and its possible gender imbalance is the key to consider OP's question on Latin literacy among women in the 11th century Scandinavia.

Compared with other areas, Scandinavia (including the Northern Isles like Iceland) can be characterized as very weak presence of monastic institutions in its early phase of Christianization. To give some examples, Benedictine (male) monasteries Þingeyrar and Munkaþvera, famous as literature production centers in the late 12th and early 13th century, founded only respectively in 1133 and 1155 CE (Johansson in Bisgaard & Nyberg red. 2006: 159). It means that even male (and elite) Icelanders did not always enjoy good access to the education of Latin alphabets in the 11th century. Before that, influential Icelanders had relied mainly on one of the two options:

• To send their son to study abroad: Monasteries in Germany and cathedral schools in England are mentioned in the 12th century.
• instead, ask the local Icelandic bishop or his cathedral school to educate their son for future priesthood, as narrated in Chapter 9 of the Book of the Icelanders (Grønlie trans. 2006: 10).

Since the oldest foundation of nunnery in Iceland dates back only to 1186 CE (Benedictine Nunnery Kirkjubær), women certainly apparently had less option in the 12th century, but in the first place we don't much about the literacy except for the very limited number of Scandinavians (even the royal members) in the 11th century.

[Continued to be part 2]