Interesting question! To get it out of the way, the Go-Hōjō (Latter Hōjō) / Sengoku Hōjō / Odawara Hōjō etc (I will refer to them as Go-Hōjō from this point) were not directly related by blood to the Hōjō of the Kamakura period. Hōjō Ujitsuna was the first of the Go-Hōjō to take the name Hōjō. Prior to this he was Ise Ujitsuna, son of Ise Shinkuro/Nagauji/Sōzui (posthumously known of Hōjō Sōun). The Ise were descendants of the Taira (and more specifically, the Kammu Heishi branch) as were the Hōjō of Kamakura, but they were not directly related. The taking of a discontinued house name (and thus revival of said warrior house) was not an uncommon occurrence during the Sengoku period. David Spafford wrote an excellent article in the Harvard Journal of Asiatic Studies Vol. 74 no. 2, titled ‘Whats in a Name? House Revival, Adoption, and the Bounds of Family in Late Medieval Japan’ which discusses this topic. He describes a conceptual piece of Ise Ujitsuna taking the name Hōjō when saying:

“Setting aside Hideyoshi, who after all was trying to fashion an elite pedigree from himself where there was none, both Ujitsuna and Ieyasu carried out their respective name changes in the guise of restoring their lineages’ original names. Or, to describe the processes in the terms I use in this essay: both warlords carried out operations analogous to adoptions. Members of junior lineages (themselves) were selected as heirs to the (vacant) headships of extinct but respectably pedigreed main lineages in their heraldic groups.” - page 288-289

So while the Go-Hōjō (in actuality) were not direct descendants of the Kamakura Hōjō, the act was not simply the taking of the name onto their already existing house name (Ise), but rather would be viewed as a revival of the house (Hōjō) as a whole. At which point, we can move onto your question as to why they would do this. There could be numerous theories as to why, but a prevalent one is that the Go-Hōjō looked to signify a connection with the Kamakura Hōjō, and their position within the Kantō region. The Kamakura period Hōjō had maintained control over the Kantō region for much of the Kamakura period. The Go-Hōjō (who were a rising power in the Kantō, through revival of the Hōjō name/house, looked to mark their position within the region. As David Spafford put it in his book A Sense of Place: The Political Landscape in Late Medieval Japan, “in 1524 he [Ujitsuna] conquered the castle of Edo and changed the family’s name from Ise to Hōjō , signalling his intent to rule the whole region as the original Hōjō had.” This move (name change) was not the only one made by the Go-Hōjō to tie themselves ( in name, symbolically, and conceptually) to the Hōjō of Kamakura. They also (as discussed by Michael P. Birt in his journal article Samurai in Passage: The Transformation of the Sixteenth Century Kanto) established a council, the ‘hyōjōshū’, which was a title of a major governing body from the heyday of the Kamakura period Hōjō. This is brought up to again highlight that the Go-Hōjō’s taking of the name from the earlier Hōjō was a broader visioned move than taking the name itself, but highlights both the Go-Hōjō’s self perception of the time, and their goals for the future (in terms of establishing control over the Kantō).