てでも is a combination of the conjunction-particle て, and the prefix adverbial particle でも. Regular interpretations include ‘no matter what’, or ‘even if I have to (A)’. When connected to verbs,
てでも just emphasizes that ‘even if (A) needs to be done, it will be’, and expresses the speaker’s strong determination to do something, regardless of what the consequences are, or how difficult it might be.
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あの最新（さいしん）のノートパソコンは高（たか）いお金（かね）を払（はら）ってでも手（て）に入（い）れたい。
I want that laptop, even if I have to pay a lot of money for it.
寒（さむ）い中（なか）並（なら）んででもあそこのラーメンを食（た）べてみたい。
I want to eat their ramen, even if I have to line up in the cold.
このレポートは徹夜（てつや）してでも終（お）わらせなければいけない。
I need to finish this essay, even if it means that I need to stay up all night.
Due to
てでも expressing strong will, it will often be seen in sentences that include other phrases that mark desire or intent, such as ほしい, たい, and ようと思（おも）う.
Fun-fact - Although てでも is considered a fairly slang expression due to the use of てでも rather than the usual ても that would be expected of a verb, the construction itself of て-form partnered with an adverbial particle is quite common, and often marks emphasis or comparison. This can be seen very regularly with ては in patterns like てはいけない or てはだめ.