First of all, is this really a full sentence? I googled and found this song. If your phrase is from this, 寄る辺ない侘しさに is a part of a sentence, not a full sentence. Please always provide the entire context.
This に is a cause/reason marker that is used with a word related to emotion or psychological reactions.
- に in コントロールに苦しまなくなった
- When do you use に to indicate a cause or reason for something?
- Grammar of nounの多さに
- Meaning of に in この狭さに
侘しい is an i-adjective, but note that it has already been nominalized with さ ("-ness") here. 侘しさ is a noun meaning "loneliness" or "melancholy".
寄る辺ない侘しさに頬杖ついてるようじゃ変わらない
(literally) If it's like you rest your cheek in your hand because of desolation with nowhere to turn, things won't change.
→ When you feel desolate with nowhere to turn, just resting your cheek in your hand won't change anything.
Even if your 寄る辺ない侘しさに is not from this song, に is still a cause/reason marker, so it's still something along the lines of "because of". If this is a book title, it indicates this person psychologically reacted to the 侘しさ in some way or another, so it might be translated like "Loneliness Makes Me..." or "In Desolate Solitude...".