Is there a prayer that is said by others on behalf of someone who is about to undergo a dangerous (non-cosmetic) surgery? Is the מי שברך the most appropriate?
The gemara says (Brachos 60a, on the bottom):
אמר רב אחא הנכנס להקיז דם אומר יהי רצון מלפניך יי' אלקי שיהא עסק זה לי לרפואה ותרפאני כי אל רופא נאמן אתה ורפואתך אמת לפי שאין דרכן של בני אדם לרפאות אלא שנהגו
Rav Acha said that someone who goes for bloodletting should say "May it be Your Will, Hashem my God, that this practice should be a healing, and I should be healed; for You are a trustworthy Healer and your healing is true," because people don't get healed by themselves (they need God to heal them).
( free translation )
הַנִּכְנָס לְהַקִּיז דָּם אוֹמֵר יְהִי רָצוֹן מִלְּפָנֶיךָ יְיָ' אֱלֹהַי שֶׁיְּהֵא עֵסֶק זֶה לִי לִרְפוּאָה כִּי רוֹפֵא חִנָּם אָתָּה. וּכְשֶׁיֵּצֵא אוֹמֵר בָּרוּךְ אַתָּה יְיָ' רוֹפֵא חוֹלִים
One who goes for bloodletting should say "May it be Your Will, Hashem my God, that this practice should be a healing for me, because you are a free Doctor." When he is finished, he should say "Blessed are You, O God, Healer of the sick."
( free translation )
I've seen some cite the above Shulchan Aruch to require saying this before taking any medicine1 (I do this without using שם השם, when I remember to), even though he only said it regarding bloodletting. I would argue that bloodletting is close enough to surgery so that the same prayer can be used.
1 Some have even made this into a card to hand out, as mentioned in one of the comments to the question.
I would recommend Refuah Shlema. It comes from Numbers 12:13 when Moses asked God to heal Miriam, "Please God, heal her." From my understanding, our sages have taught us that this did not ask for complete healing, but for simple healing. Thus adding the adjective "shlema," asks for a prayer of healing to the greatest extent.
Mi-sheberakh avoteinu v'imoteinu, Avraham v'Sarah, Yitzhak v'Rivkah, Ya'akov, Rachel v'Leah hu y'varekh et (recite the English and Hebrew name) v'yavi aleihem refuat hanefesh u'refuat haguf yachad im kol cholei amo Yisrael. Barukh atah Adonai, rofeh ha'cholim.
May the One who was a source of blessing for our ancestors, bring blessings of healing upon (recite the English and Hebrew name), a healing of body and a healing of spirit. May those in whose care they are entrusted, be gifted with wisdom and skill, and those who surround them, be gifted with love and trust, openness and support in their care. And may they be healed along with all those who are in need. Blessed are You, Source of healing. Amen.