Costolo has faced increased criticism as Twitter's stock has fallen 23 percent in the last three months, partly fueled by disappointing first-quarter results. At the Re/code Code Conference on Thursday, Costolo maintained that he and Twitter's board are "totally in sync."
Kelly believes that either Costolo will turn the company around or someone will replace him, both of which bode well for the stock.
"You can buy Twitter on pure speculation that he's going to be gone," he said.
Twitter shares rose more than 1 percent Thursday, closing just below $38.
Google rolled out new technology at its annual developers' conference on Thursday. The Internet and technology giant unveiled Android Pay, a mobile payment system, and a photo sharing app, among other developments.
Google has "a lot of things to like," said trader Karen Finerman, whose firm, Metropolitan Capital Advisors, owns Google stock. She called the company "a cash machine."
"The thing that I like the most is the valuation," she said.
Trader Steve Grasso added that he recently sold his Google stake but was considering taking a new position. Google Class A stock closed Thursday barely lower around $554 per share.
Wearable camera maker GoPro also made waves at the Code Conference, unveiling plans to enter the virtual reality and drone market. GoPro looks to be diversifying and reducing its reliance on its core product, Kelly said.
"You buy GoPro because [CEO] Nick Woodman can get content nobody else is going to get," he said.
GoPro shares closed more than 6 percent higher Thursday.
Dan is long GOOGL June call fly, TWTR, BBRY June calls, TLT June call butterfly, SO, DE June put fly and INTC July put spread. He is short SO Aug calls.
Steve Grasso is long AAPL, BAC, DD, EVGN, MJNA, PFE, T, TWTR and GDX. His firm is long TWTR, AMZN and MCD. His kids own EFG, EFA, EWJ, IJR and SPY.
Brian Kelly is long DXGE, BTC=, BBRY and U.S. dollar. He is short Australian dollar, Canadian dollar, euro, yen, yuan and WTI crude oil.
Karen is long BABA, BAC, C, FINL, FL, GOOG, GOOGL, JPM, M, KORS and SEDG. She is short SPY. Her firm is long ANTM, AAPL, BAC, C, DIS, FBT, FINL, FL, GOOG, GOOGL, GPS, IBB, JPM, M, KORS, SUNE, URI, XBI and SPY puts. Her firm is short IWM, SPY and MDY. Karen Finerman is on the board of GrafTech International.