Rav Sa'adya Gaon writes in Emunot V'deot (Ma'amar 2 s.v. v'hinei ani) that in reference to God, remembrance refers to salvation:
ובהצלת הברואים מענין שמצער אותם קוראים אותו זכירה, שאמרו (בראשית ח' א') ויזכר אלהים את נח. (שם ל' כ"ב) ויזכר אלהים את רחל והדומה לזה
And saving creatures from something that afflicts them, this is called "remembrance". As in his statement: "And God remembered Noah" (Gen. 8:1), "and God remembered Rachel" (Gen. 30:22).
R. Elijah Mizrahi similarly writes (Gen 19:29 s.v. k'shehaya) that remembering refers to saving. See also Hovot HaLevavot (Sha'ar HaYihud: 10).
Ibn Ezra to that verse (Genesis 8:1) explains that certainly God never forgets anything. Rather, it is a figure of speech indicating God acting based on a previous event:
ויזכור אלהים - חלילה להיות שכחה לקטון שבמלאכי השרת, אף כי ליוצר הכל, שאין שכחה נגדו... רק כאשר תראה טובת השם בארץ ידבר הכתוב, כי זכר אשר הטיב לו בא לפני השם.
And God remembered: heaven fore-fend that forgetfulness apply to the least of [God's] ministering angels, let alone to the Creator of everything, for there is no forgetfulness that can apply to him...Just that when God's goodness is manifested in the world, the verse will describe it as a remembrance of goodness done to God, coming to him.
Radak writes very similarly (ibid). He applies the principle that "the Torah speaks in the language of Man", (cf. Berakhot 31b, Moreh Nevukhim 1:29) to explain that the Torah uses terminology familiar to humans in reference to God, even though their most literal meaning doesn't apply to him. He uses this to similarly explain "But I will for their sakes remember the covenant of their ancestors" (Leviticus 26:45), and related verses.
This 18th century commentary of R. Naftali Hertz Wessley cites support for such explanations from Genesis (40:29) "Yet the chief butler did not remember Joseph, but forgot him". Certainly, he asserts, the butler often remembered his time in prison and his experience with Joseph there. The term "remembering" in that context, therefore refers to focusing on something's characteristics, in order to respond positively or negatively to something. The butler ignored Joseph, so he is said to have "forgotten". Similarly, when God is said to forget or remember, it refers to his behaviour.
Similarly, note Lamentations (5:20):
לָמָּה לָנֶצַח תִּשְׁכָּחֵנוּ, תַּעַזְבֵנוּ לְאֹרֶךְ יָמִים
Why do you forget us forever; forsake us for the length of days.
The structure of the verse parallels לָנֶצַח with לְאֹרֶךְ יָמִים (terms for eternity), and תִּשְׁכָּחֵנוּ (forgetting us) with תַּעַזְבֵנוּ (abandoning us).