[REVISED] There is a machloket among the Tosafists about whether if you forget yaale v'yavo on mincha of rosh chodesh you should say two tefilot in the evening to make up for it. The Rabbis of Provence say you missed a prayer so you need two the next time; the Ri says you do not gain anything by praying a second time because you cannot correct what you missed since we no longer mention rosh chodesh at night. See MB 108:10. Because it is a safek, the Shulchan Aruch concludes one should pray it as a tefilat nedavah (voluntary prayer). SA 108:10; MB 108:32-33. There is also a machloket rishonim whether one can make a voluntary tefilah on shabat. The SA and MB conclude you should not. SA 107:1; MB 108:33.
So the big question is whether messing up the prayer for rain is similar to forgetting yaaleh v'yavo or different. This appears to be a machloket acharonim. The Magen Avraham (end of 108) writes that for sure one would repeat a second prayer if one forgot yaaleh v'yavo on the first day of yom tov [because yaaleh v'yavo is also relevant at night] and then says: וה"ה אם שכח ותן טל ומטר או משיב הרוח and so too if one forgot tal u'matar or mashiv haruach. The MB 108:35 also says יעלה ויבא - בדיוקא נקט יעלה ויבא דאם שכח טל ומטר או משיב הרוח בשבת ור"ח במנחה או ששכח שאר דבר המצריכו לחזור ולהתפלל לכו"ע יתפלל בערב שתים דהא ירויח זה בתפילת התשלומין: the rule is specific to yaaleh v'yavo, but if he forgot tal u'matar or mashiv haruach on shabat [erev shabat?] or rosh chodesh at mincha or he forgot another thing that he must repeat his prayer for, then according to everyone he should pray two at night... Both of these sources seem to be saying that there is a difference between yaaleh v'yavo and an error in one of the brachot themselves. Rav Chaim Soloveichik says that explicitly in his "stencil" chiddushim.: according to the Ri, omitting yaaleh v'yavo on rosh chodesh is a valid prayer but fails to include a necessary addition, so you only need to pray again so you can mention yaaleh v'yavo; but erring in the blessing for rain invalidates the whole tefilah and it's as if you did not pray mincha at all, so you need to pray tashlumin. (HT @Joel K). However, this contemporary article from "Beineinu" reads the Magen Avraham and Mishnah Berurah exactly the opposite: that all errors are the same. This online teshuvah from Rav Yitzchak ben Yosef also treats the two cases as the same and says you should not say tashlumin Friday night if you forgot to ask for rain Friday afternoon.
A second question is whether incorrectly asking for rain in the summer is the same or different from omitting the prayer for rain in the winter. The Shulchan Aruch 117:3 says that if you mention rain when you shouldn't, you must repeat the shmoneh esrei as an actual obligation, unless you live in a country that needs rain in the summer in which case it is a safek whether you need to repeat and you should add a second tefilah as a voluntary offering (nedavah). SA 117:2. In other words, in the default case, incorrectly praying for rain invalidates the whole tefilah. Thus, according to the Grach, and my understanding of the Magen Avraham and Mishnah Berurah, you should pray tashlumin on Friday night. (If forgetting to mention rain is seen as "not praying" rather than omitting an addition like yaaleh v'yavo, kal v'chomer (a fortiori) improperly mentioning rain should have the same status. If all you had to do was "not ask for rain," you could do that without repeating the shmoneh esrei at all!)
According to the opposing approach--that you should not do tashlumin unless you can fix your mistake by doing so--I could see an argument that the obligation to not pray for rain is common between Friday afternoon and Shabat, so even the Ri would agree you can do tashlumin. In which case there would be no safek and tashlumin would be proper even on shabat. But I don't think this is correct, because at the end of the day you messed up a specific blessing (the blessing of years), and you will not be saying that blessing at night. So I would think that the rule of not mentioning rain should be basically the same as the rule of mentioning it, but tzarikh iyun.
The Beineunu article I linked above proposes another creative solution. Technically, a prayer on shabat that mentions the weekday blessings is valid b'dieved. SA 268:2. So in your case, you could just include the additional weekday blessings you messed up on Friday night and conclude with the blessing for shabat and that would even satisfy the opinion of the Ri. However the article itself leaves this as a matter of tzarikh iyun (requiring further investigation).
Finally note that, in places that need rain in the summer, the tashlumin for mentioning tal u'matar in the summer is a nedava anyway because there is an independent safek about whether you need to repeat at all, so the rule not to make voluntary prayers on shabat would apply regardless. The poskim mention Spain, Ashkenaz, and Eastern Europe as being in this category. See Biur Halacha to 117:2. It seems to me that much of the U.S. has a similar climate to Ashkenaz in that farmers rely on rain in at least part of the time between Pesach and Sukkot. I have not found a halakhic source discussing hte matter, but I did ask a posek in Lakewood and he thought that svara comparing U.S. and Europe was "pashut." However, there are other opinions that say these countries only count as "needing rain" in a time of drought, and not times of summer when they need and are receiving rain. Biur Halachah (id.) brings both opinions and concludes that since it is a safek one should repeat the tefilah conditionally. Similar issues also arise in the Southern Hemisphere where the summer and winter can be the opposite of what they are in Israel. Where they apply, these considerations add additional variables that could make repeating the shmone esrei a safek nedava and not a vadai tashlumin, suggesting it should not be made on shabat.
TL;DR if one forgot yaaleh v'yavo on the afternoon of Friday Rosh Chodesh and shabat is not Rosh Chodesh, the SA and MB rule one should not say two tefilot on shabat because you will not be saying yaaleh v'yavo anyway, which makes tashlumin a safek and we do not do safek tashlumin on shabat . There is a machloket acharonim if this rule is specific to yaaleh v'yavo or also applies to defects in the blessings themselves. It seems to me that omitting the prayer for rain in winter and including it in summer should have the same status, though I concede that is not 100% clear. In addition, if you live in a place where rain is needed in the summer there is also a separate safek whether you need tashlumin at all, which would be an additional factor against praying twice Friday night.
